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Gravitaxis in motile micro-organisms: The role of fore-aft body asymmetry
Abstract
Scale model experiments on axially symmetric bodies exhibiting fore–aft asymmetry
are described. Body shapes are specified by a three parameter equation: two of the
parameters (a and b) describe the length and breadth of the body and the third (c)
the degree of asymmetry. Objects of this shape orientate as they sediment downwards
under gravity until the narrower end lies uppermost, after which they fall vertically
downward with no further change in orientation. For the range of parameters
investigated the sedimentation velocities, both when vertical and horizontal, are governed
principally by a and b, while the rate of orientation is determined by c. The
sedimentation characteristics of bodies which cannot be described exactly by the equation
can be predicted approximately using best-fit values for a, b and c. These results
are applied to consider the role of front–rear asymmetry in ciliated free-swimming
micro-organisms. The shape asymmetry is probably sufficient to account for the observed
orientation rates in the ciliated protozoan Paramecium. It is suggested that
these results may be used to deduce the sedimentation behaviour of ciliates from
microscope images of individual cells. In small flagellates such as Chlamydomonas the
orientating effects of the protruding flagella are much larger than the effects of cell
body asymmetry. The extreme sensitivity of the orientation rate to slight changes in
body shape and flagellar beat patterns may explain why experiments to distinguish
between various orientational mechanisms involved in gravitaxis have in the past
produced equivocal results.
... Assuming a body of revolution swimming in a fluid at a low Reynolds number, the translational and rotational equations of motion are decoupled [49,71,89,90], allowing us to write the following equations along the major-and the minor-axis ( Figure 6, under force-free conditions: ...
... Changes in both the cell density and cell shape can impact orientational stability of swimming cells [23,89]. For instance, when a cell expels water or gas molecules by regulating intracellular vacuoles, both the density distribution within the cell and the cell shape undergo considerable change. ...
... While the sedimentation rate of symmetrically-shaped prolate spheroids (which are denser than the surrounding fluid) can be estimated analytically [92], the calculations for distorted prolate spheroids with fore-aft asymmetry are more involved. One needs to account for the additional reorientation term, since asymmetric prolate sheroids rotate while sedimenting downwards at instantaneous rates given by the degree of asymmetry [23,71,89,92]. The sedimentation characteristics of an asymmetric cell body, with rotational symmetry, can be derived using the dumb-bell model described by Happel and Brenner [92], which approximates the asymmetric fore-aft to be constructed equivalently out of two spherical blobs of different radii, connected by a light, rigid inextensible rod. ...
Planktonic active matter represents an emergent system spanning different scales: individual, population and community; and complexity arising from sub-cellular and cellular to collective and ecosystem scale dynamics. This cross-scale active matter system responds to a range of abiotic (temperature, fluid flow and light conditions) and biotic factors (nutrients, pH, secondary metabolites) characteristic to the relevant ecosystems they are part of. Active modulation of cell phenotypes, including morphology, motility, and intracellular organization enable planktonic microbes to dynamically interact with other individuals and species; and adapt - often rapidly - to the changes in their environment. In this chapter, I discuss both traditional and contemporary approaches to study the dynamics of this multi-scale active matter system from a mechanistic standpoint, with specific references to their local settings and their ability to actively tune the behaviour and physiology, and the emergent structures and functions they elicit under natural ecological constraints as well as due to the shifting climatic trends.
... While in some organisms such an alignment might be dominated by an active physiological mechanism [4], it has also been suggested that the alignment relative to the gravitational force is the result of an inhomogeneous mass density within the organism (bottom-heaviness) similar to a buoy in the ocean. Indeed, this effect is employed by certain phytoplankton species [5], Chlamydomonas [6], or pluteus larvae of the sea urchin [7]. Interestingly, sedimentation experiments with Paramecium, and gastrula larvae [7] also provide evidence for another mechanism responsible for gravitaxis which results from a fore-rear asymmetry in the organisms' shape. ...
... This suggests that the alignment cannot be attributed to an inhomogeneous mass distribution. The observed alignment with the broader end pointing downwards is characteristic for sedimenting objects with homogeneous mass distribution and a front-rear asymmetry [6]. This can easily be understood by considering an asymmetric dumbbell formed by two spheres with identical mass density but different radii R 1 and R 2 > R 1 . ...
... The sedimentation speed of a single sphere due to gravitational and viscous forces is v ∝ R 2 . Therefore, if hydrodynamic interactions between the spheres are ignored, the dumbbell experiences a viscous torque resulting in an alignment where the bigger sphere is below the smaller one [6]. ...
Many motile microorganisms adjust their swimming motion relative to the gravitational field and thus counteract sedimentation to the ground. This gravitactic behaviour is often the result of an inhomogeneous mass distribution, which aligns the microorganism similar to a buoy. However, it has been suggested that gravitaxis can also result from a geometric fore-rear asymmetry, typical for many self-propelling organisms. Despite several attempts, no conclusive evidence for such an asymmetry-induced gravitactic motion exists. Here, we study the motion of asymmetric self-propelled colloidal particles which have a homogeneous mass density and a well-defined shape. In experiments and by theoretical modelling, we demonstrate that a shape anisotropy alone is sufficient to induce gravitactic motion with either preferential upward or downward swimming. In addition, also trochoid-like trajectories transversal to the direction of gravity are observed.
... We shall discard the c 0 term, since eventually we shall require the response function R to have zero mean, so that an additive constant to c in (2) has no effect on ∆ . Substituting (13) into (11) and discarding terms containing c 0 , one finds ...
... For instance, Salmonella typhimurium is a denser-than-water bacterium that performs run and tumble chemotaxis in a similar manner to E. coli, and mutant non-tumbling S. typhimurium cells swim downward on average and collect at the bottom of the vessel in which they are contained [1]. This is presumably because the hydrodynamic drag on the flagella is larger than that on the cell body, so a torque is exerted on a non-vertically-oriented cell [11]. The implication is that it would be disadvantageous for a cell to have α p very close to 1 since it would then swim downward on average, regardless of the direction of the chemoattractant gradient. ...
Escherichia coli is a motile bacterium that moves up a chemoattractant gradient by performing a biased random walk composed of alternating runs and tumbles. Previous models of run and tumble chemotaxis neglect one or more features of the motion, namely (i) a cell cannot directly detect a chemoattractant gradient but rather makes temporal comparisons of chemoattractant concentration, (ii) rather than being entirely random, tumbles exhibit persistence of direction, meaning that the new direction after a tumble is more likely to be in the forward hemisphere, and (iii) rotational Brownian motion makes it impossible for an E. coli cell to swim in a straight line during a run. This paper presents an analytic calculation of the chemotactic drift velocity taking account of (i), (ii) and (iii), for weak chemotaxis. The analytic results are verified by Monte Carlo simulation. The results reveal a synergy between temporal comparisons and persistence that enhances the drift velocity, while rotational Brownian motion reduces the drift velocity.
... Numerical simulations of our system lend insight into the properties of these flux loops. We employ a minimal mechanistic model of C. reinhardtii cells as asymmetric dumbbells (SI Appendix, Fig. S1), representing the characteristic fore-aft asymmetry of the cell's body and of a larger space spanned by the stroke-averaged flagella (30,38,39). Note that other microswimmer geometries produce fluxes as well, though without matching experiments quantitatively (SI Appendix, section V). ...
... Numerical Model and Simulation Parameters The C. reinhardtii cells are modeled as asymmetric dumbbells (SI Appendix, Fig. S1) with a large sphere in front and a smaller sphere in the back, representing the fore-aft asymmetry of body and appendages (38,39). The equation of motion for the position r of the active dumbbell is given by dr dt ...
Significance
Motile microorganisms commonly live in porous media comprising microhabitats filled with interfaces of complex shape. On such small scales, the interactions with these interfaces, rather than external gradients, dominate their motion in the search for favorable living conditions. We demonstrate with experiments and theory that the geometry of confining interfaces shapes the topology of the most likely, average trajectory, leading to directed fluxes of probability that are not exclusively localized at the near-wall region. Employing this principle allows us to actively shape a microbe’s average direction of movement, which could be of use in the design of topological transport mechanisms for microfluidic environments.
... Numerical simulations of our system lend insight into the properties of these flux loops. We employ a minimal mechanistic model of C. reinhardtii cells as asymmetric dumbbells (see Fig. S1), representing the fore-aft asymmetry of body and appendages 28,33,34 . The translational dynamics of the dumbbell are governed by an overdamped Langevin equation: dr dt = v 0 e + µ w F w + η, where r is the position of the geometric center the dumbbell, v 0 is the self-propulsion velocity, e is the orientation of the dumbbell, µ w is the mobility and F w is the force stemming from steric wall interactions. ...
... The C. reinhardtii cells are modelled as asymmetric dumbbells (see Fig. S1), representing the fore-aft asymmetry of body and appendages 33,34 . The equation of motion for the position r of the active dumbbell is given by ...
When the motion of a motile cell is observed closely, it appears erratic, and yet the combination of nonequilibrium forces and surfaces can produce striking examples of collective organization in microbial systems. While our current understanding is based on bulk systems or idealized geometries, it is not clear at which length scale self-organization emerges. Here, using experiments, analytical and numerical calculations we study the motion of motile cells under controlled microfluidic conditions, and demonstrate that a robust topology of probability flux loops organizes active motion even at the level of a single cell exploring an isolated habitat. By accounting for the interplay of activity and interfacial forces, we find that the boundary's curvature determines the nonequilibrium probability fluxes of the motion, which can be controlled directly. We theoretically predict a universal relation between fluxes and global geometric properties that is directly confirmed by experiments. Our findings open the possibility to decipher the most probable trajectories of motile cells and may enable the design of active topological materials.
... Für die vertikale Orientierung des Bakteriums kommen als Ursache ein Dichtegradient innerhalb des Organismus oder anisotrope hydrodynamische Eekte aufgrund der asymmetrischen Form in Frage. Welcher Mechanismus für die negative Gravitaxis ausschlaggebend ist, ist noch nicht abschlieÿend geklärt, eine Übersicht der experimentellen Ergebnisse ndet sich in[162]. Viele Mikroorganismen besitzen eine Form mit Vorne-Hinten-Asymmetrie ('Fore-Aft Asymmetry'), d.h. ...
... der hintere Teil, an dem der Antrieb sitzt, ist deutlich breiter als die Spitze (die Form ähnelt einem Tannenzapfen). Generell orientieren sich Objekte mit Vorne-Hinten-Asymmetrie aufgrund von hydrodynamischen Wechselwirkungen mit ihrer breiteren Seite in Richtung der Gravitationskraft[162]. Dieses Verhalten kann mit einem einfachen Hantelmodell verstanden werden. ...
Reibung lässt sich auf makroskopischer Längenskala durch das Amontonsche Gesetz beschreiben, welches besagt, dass Reibungs- und Normalkraft zueinander direkt proportional sind. Dieser einfache Zusammenhang beruht auf dem Scheren unzähliger Mikrokontakte, ein Mechanismus, der erst in den 1950er Jahren theoretisch verstanden und erst nach der Jahrtausendwende experimentell aufgelöst wurde.
Um grundlegende Erkenntnisse über Reibung zu gewinnen, müssen allerdings die Mechanismen verstanden werden, die zum Brechen eines einzelnen Mikrokontakts führen, also Prozesse, die auf Längenskalen von Mikro- bis Nanometern ablaufen. Dies führte zur Entwicklung des Forschungsfelds der Nanotribologie, welches Reibung, Schmierung und Verschleiß auf der Nanoskala behandelt.
Ein wichtiges theoretisches Werkzeug der Nanotribologie sind simplifizierte tribologische Modelle, wie das Tomlinson oder das Frenkel-Kontorova (FK) Modell. Das Tomlinson Modell beschreibt punktförmige Kontakte, für realistischere, d.h. ausgedehnte Kontaktgeometrien wird von theoretischer Seite das getriebene Frenkel-Kontorova Modell verwendet. Während die Vorhersagen des Tomlinson Modells durch Messungen mit dem Rasterkraftmikroskop bestätigt wurden, existiert bisher kein experimentelles System, um das FK Modell detailliert zu studieren. Von besonderem Interesse sind hierbei sog. topologische Solitonen, die im Rahmen des Frenkel-Kontorova Modells vorhergesagt werden und welche einen effizienten Mechanismus für den Massentransport auf kleinen Längenskalen darstellen. Die gezielte Erzeugung und Manipulation topologischer Solitonen bietet eine Perspektive, Reibung auch auf der Nanoskala zu reduzieren.
In dieser Arbeit wird die erste experimentelle Realisierung eines zweidimensionalen getriebenen Frenkel-Kontorova Modells verwirklicht. Hierfür dienen kolloidale Monolagen miteinander wechselwirkender Partikel, welche über stationäre lichtinduzierte Substratpotentiale getrieben werden. Der Aufbau erweist sich dabei als ideales Modellsystem, da nahezu alle relevanten Parameter in situ variiert werden können. Die geladenen Partikel wechselwirken repulsiv mittels eines Yukawa Potentials, dessen Reichweite über die Ionenkonzentration der Lösung kontrolliert wird. Die Interferenz mehrerer Laserstrahlen erlaubt es, ausgedehnte Lichtfelder mit mehreren zehntausend Minima zu erzeugen, welche als Substratpotential für die kolloidale Monolage dienen. Durch Änderung von Anzahl und Anordnung der Strahlen können sowohl periodische, als auch quasiperiodische Substratpotentiale generiert werden, deren Längenskalen durch Änderung des Einfallswinkels der Strahlen variiert werden können.
... Rotation impacts the behavior of individual particles through its effects on drag (Clift et al. 2005;Mortensen et al. 2008b), as well as navigation and perception. For example, rotation may interfere with gravitaxis and directed swimming of plankton (Machemer and Bräucker 1992;Roberts and Deacon 2002). Two-particle interactions, such as collision and avoidance, are dependent on rotation and orientation (Koch and Shaqfeh 1989). ...
... The behavior of entire particle communities may also be affected by rotation, for example, by controlling preferential alignment. Particle alignment can be due to settling or shear (Roberts and Deacon 2002;Reidenbach et al. 2009); rotational diffusion acts as a competing mechanism that works against alignment. The interplay among these mechanisms can cause unique visual signatures as the alignment evolves, due to alignment's effects on the optical properties of a suspension. ...
Through laboratory measurements, we compared the rotation of spherical and ellipsoidal particles in homogeneous, isotropic turbulence. We found that the particles' angular velocity statistics are well described by an Ornstein– Uhlenbeck (OU) process. This theoretical model predicts that the Lagrangian autocovariance of particles' angular velocity will decay exponentially. We measured the autocovariance by using stereoscopic particle image velocimetry (SPIV) applied to spherical and ellipsoidal particles whose size was within the inertial subrange of the ambient turbulence. SPIV resolves the motion of points interior to the particles, from which we calculated the solid body rotation of the particles. This provided us with the angular velocity time series for individual particles. Through ensemble statistics, we determined the autocovariance of angular velocity and confirmed that it matches the form predicted by an OU process. We found that in this stochastic framework the autocovariances of both the ellipsoids and spheres are statistically identical, suggesting that rotation is controlled by the large scales of turbulence. We can further use the autocovariance curve to quantify the turbulent rotational diffusivity and discuss its implications for the transport of aquatic organisms in natural turbulence.
... However, as the topic is generally well described (see Refs. [104,105] as well as the comprehensive introductions in Refs. [106,107]), I restrict myself to a brief overview here. ...
... The asymmetrical shape of the organism may play an important role in the gravitational response of ciliates, and even slight changes in shape may have a profound effect on the orientation of the organism [107]. The prevailing theory today, however, is that at least some protozoa are capable of sensing gravity directly by the cytoplasmic contents of the cell exerting a pressure on the lower membrane, thus activating stretch-sensitive calcium-specific ion channels [105]. ...
... It was proposed that gravitaxis is mediated by passive physical forces, such as buoyancy based on a fore/aft asymmetry (Brinkmann 1968;Roberts 1970). Passive gravitactic alignment of the cells can be explained by a gravity-buoyancy model or a gravity-drag model (Roberts 1970;Machemer et al., 1992;Mogami et al., 2001;Roberts et al., 2002). In both models the center of mass is different from the geometrical center of the cell (centroid). ...
... The gravity-drag model proposes a fore-aft asymmetry of the cell shape (e.g., rear end thicker than the front end). Due to differences in the Reynolds numbers (fluid mechanic of small particles) of the thicker rear end compared to the thinner front end the rear end sinks faster (Roberts et al., 2002). Later results revealed that passive alignment probably plays a role in gravitaxis, above all in stationary cultures, but intensive research on gravitaxis revealed that this phenomenon is to a large extend based on physiological orientation mechanism. ...
This article provides an overview of the research of phototaxis and gravitaxis in the unicellular
flagellate Euglena gracilis. The cells show positive phototaxis at low light intensities (swimming
towards the light source, below 10 W/m2, sunlight) and a negative one upon high irradiances
(>10 W/m2). Phototaxis is based on a blue light-activated adenylyl cyclase, which produces
cAMP upon irradiation. The further steps following the transient rise of the cAMP concentration
are still unknown. Gravitaxis is a movement parallel to an acceleration vector. In the absence
of external stimuli the cells swim upward in the water column (negative gravitaxis), upon
stress gravitaxis inverts into a positive one. The results of sounding rocket campaigns and
of a large number of ground experiments led to the following model of signal perception and
transduction in gravitaxis of Euglena gracilis: The body of the cell is heavier than the
surrounding medium, sediments and thereby exerts a force onto the lower membrane. Upon
deviation from a vertical swimming path mechano-sensitive elements are activated. This leads
to an increase in the intracellular calcium concentration and a change of the membrane
potential. The increased calcium concentration regulates the action of calmodulin, which in
turn modulates an adenylyl cyclase, which converts ATP to cAMP. cAMP probably controls
the flagellar beat and/or probably the shape of the cell. Phototaxis as well as gravitaxis are
probably controlled by certain protein kinases. The blockage of different protein kinases (PK)
by means of RNAi gave rise to the assumption that both signal pathways activate certain
PKs. The blockage of one PK resulted in a loss of gravitaxis, the blockage of another one in
loss of phototactic behaviour.
... This phenomenon is not observed in nonswimming Paramecium (11). On the other hand, there also exist passive responses for which a physical field such as gravity may directly orient an organism without eliciting a physiological change (13)(14)(15). Although these responses are simpler, they can still provide survival advantages. ...
... In our model of the magnetic reorientation of motile Paramecium, we have been able to neglect the influence of gravitational torques. Gravitational torques can arise because of shape asymmetry (14) and/or internal density inhomogeneities and tend to orient the Paramecium parallel with the gravity vector. In our experiments, the gravity vector and magnetic field vector are parallel. ...
Recent experiments on cell division suggest the application of intense
static magnetic fields as a novel tool for the manipulation of
biological systems [1]. The magnetic field appears to couple to the
intrinsic anisotropies in the diamagnetic components of the cells. Here,
we present measurements of the intrinsic average diamagnetic anisotropy
of the whole single celled ciliate, Paramecium Caudatum. Magnetic
fields, 2.5 T < B < 8 T were applied to immobilized
(non-swimming) Paramecium Caudatum that were suspended in a density
matched medium. The organisms align with their long axis parallel to the
applied magnetic field. Their intrinsic diamagnetic anisotropy is
3x10-11 in cgs units. We will discuss the implications of
these results for employing magnetic fields to probe the behavior of
swimming Paramecium. [1] J. M. Valles, Jr. et al., Expt. Cell Res.274,
112-118 (2002).
... The term "bottom-heaviness" refers to the inhomogeneous density distribution within a cell, with the chloroplast at the rear of the cell being denser, causing a gravitational torque that aligns the cells against gravity (26). Similarly, the "shape asymmetry" refers to the asymmetric drag force due to the presence of the flagella at the anterior of the cells, causing a similar torque during cell sedimentation (26,27). Due to negative gravitaxis, the cells tend to form a dense layer at the top surface, with the vertical cell density to theoretically follow an exponential decrease toward deeper regions (28). ...
Photosynthetic microbes have evolved and successfully adapted to the ever-changing environmental conditions in complex microhabitats throughout almost all ecosystems on Earth. In the absence of light, they can sustain their biological functionalities through aerobic respiration, and even in anoxic conditions through anaerobic metabolic activity. For a suspension of photosynthetic microbes in an anaerobic environment, individual cellular motility is directly controlled by its photosynthetic activity, i.e. the intensity of the incident light absorbed by chlorophyll. The effects of the metabolic activity on the collective motility on the population level, however, remain elusive so far. Here, we demonstrate that at high light intensities, a suspension of photosynthetically active microbes exhibits a stable reverse sedimentation profile of the cell density due to the microbes’ natural bias to move against gravity. With decreasing photosynthetic activity, and therefore suppressed individual motility, the living suspension becomes unstable giving rise to coherent bioconvective flows. The collective motility is fully reversible and manifests as regular, three-dimensional plume structures, in which flow rates and cell distributions are directly controlled via the light intensity. The coherent flows emerge in the highly unfavorable condition of lacking both light and oxygen and, thus, might help the microbial collective to expand the exploration of their natural habitat in search for better survival conditions.
... While the mechanism of some of these responses, such as chemotaxis, is active, involving the ability to sense gradients and signals, others are passive, purely resulting from external forces and torques. Gravitaxis in Paramecium, for example, is thought to arise solely due to the organisms fore-rear asymmetry [1]. Recently, there has been substantial effort to develop artificial micro-swimmers which mimic their natural counterparts in many ways [2]. ...
For natural microswimmers, the interplay of swimming activity and external flow can promote robust motion, e.g. propulsion against ("upstream rheotaxis") or perpendicular to the direction of flow. These effects are generally attributed to their complex body shapes and flagellar beat patterns. Here, using catalytic Janus particles as a model experimental system, we report on a strong directional response that occurs for spherical active particles in a channel flow. The particles align their propulsion axes to be nearly perpendicular to both the direction of flow and the normal vector of a nearby bounding surface. We develop a deterministic theoretical model of spherical microswimmers near a planar wall that captures the experimental observations. We show how the directional response emerges from the interplay of shear flow and near-surface swimming activity. Finally, adding the effect of thermal noise, we obtain probability distributions for the swimmer orientation that semi-quantitatively agree with the experimental distributions.
... The term 'bottom-heaviness' refers to the inhomogeneous density distribution within a cell, with the chloroplast at the rear of the cell being denser, causing a gravitational torque that aligns the cells against gravity [26]. Similarly, the 'shape asymmetry' refers to the asymmetric drag force due to the presence of the flagella at the anterior of the cells, causing a similar torque during cell sedimentation [26,27]. Due to negative gravitaxis, the cells tend to form a dense layer at the top surface, with the vertical cell density to theoretically follow an exponential decrease toward deeper regions [28]. ...
Photosynthetic microbes have evolved and successfully adapted to the ever-changing environmental conditions in complex microhabitats throughout almost all ecosystems on Earth. In the absence of light, they can sustain their biological functionalities through aerobic respiration, and even in anoxic conditions through anaerobic metabolic activity. For a suspension of photosynthetic microbes in an anaerobic environment, individual cellular motility is directly controlled by its photosynthetic activity, i.e. the intensity of the incident light absorbed by chlorophyll. The effects of the metabolic activity on the collective motility on the population level, however, remain elusive so far. Here, we demonstrate that at high light intensities, a suspension of photosynthetically active microbes exhibits a stable reverse sedimentation profile of the cell density due to the microbes' natural bias to move against gravity. With decreasing photosynthetic activity, and therefore suppressed individual motility, the living suspension becomes unstable giving rise to coherent bioconvective flows. The collective motility is fully reversible and manifests as regular, three-dimensional plume structures, in which flow rates and cell distributions are directly controlled via the light intensity. The coherent flows emerge in the highly unfavourable condition of lacking both light and oxygen and, thus, might help the microbial collective to expand the exploration of their natural habitat in search for better survival conditions.
... The CR cells are modelled as asymmetric dumbbells with a large sphere in front and a smaller sphere in the back, which capture in a minimal way the fore-aft asymmetry of body and appendages (Roberts and Deacon, 2002;Roberts, 2006, see Appendix 1- figure 6). The dumbbell's spheres have different radii a 1/2 and are separated by a distance ⃗ l . ...
The movement trajectories of organisms serve as dynamic read-outs of their behaviour and physiology. For microorganisms this can be difficult to resolve due to their small size and fast movement. Here, we devise a novel droplet microfluidics assay to encapsulate single micron-sized algae inside closed arenas, enabling ultralong high-speed tracking of the same cell. Comparing two model species - Chlamydomonas reinhardtii (freshwater, 2 cilia), and Pyramimonas octopus (marine, 8 cilia), we detail their highly-stereotyped yet contrasting swimming behaviours and environmental interactions. By measuring the rates and probabilities with which cells transition between a trio of motility states (smooth-forward swimming, quiescence, tumbling or excitable backward swimming), we reconstruct the control network that underlies this gait switching dynamics. A simplified model of cell-roaming in circular confinement reproduces the observed long-term behaviours and spatial fluxes, including novel boundary circulation behaviour. Finally, we establish an assay in which pairs of droplets are fused on demand, one containing a trapped cell with another containing a chemical that perturbs cellular excitability, to reveal how aneural microorganisms adapt their locomotor patterns in real-time.
... These bottom-heavy swimmers experience a gravity torque that tends to align against the direction of gravity, so that they swim upwards (Kessler 1985;Durham et al. 2013;Gustavsson et al. 2016). Also density or shape asymmetries give rise to torques in the Stokes approximation that can change the swimming direction (Roberts 1970;Jonsson 1989;Roberts & Deacon 2002;Roy et al. 2019). ...
A small spheroid settling in a quiescent fluid experiences an inertial torque that aligns it so that it settles with its broad side first. Here we show that an active particle experiences such a torque too, as it settles in a fluid at rest. For a spherical squirmer, the torque is where is the swimming velocity, is the settling velocity in the Stokes approximation, and is the equivalent fluid mass. This torque aligns the swimming direction against gravity: swimming up is stable, swimming down is unstable.
... Some of the applications of using gyrotactic microorganisms are fertilisers, biofuel, and biomicrosystems. The gravitaxis is the oriented movement of microorganisms like ciliates and flagellates because of gravity which could be either negative (upward swimming) or positive (downward swimming) [5][6][7]. The oxitactic microorganisms such as Bacillus Subtilis swim because of oxygen gradients. ...
The paper reports the natural thermo-bio-convection of gyrotactic microorganisms in a square cavity with two smaller square heat sources inside the cavity. The flow is assumed to be two-dimensional, steady, buoyancy-driven, and Newtonian. The work investigates the influences of thermal Rayleigh number, bio-convection Rayleigh number, Lewis number, and Peclet number on the natural convection heat transfer, entropy generation, and micro-organism concentration. The governing equations are discreted by finite element method. The diffusion equation is used for the motion of microorganisms, and the energy equation is considered for the effects of temperature. The main finding of this study is that both thermal and bio-convection Rayleigh number improve thermo-bio-convection performance of gyrotactic microorganisms in a square cavity with two square heaters inside the cavity. For high thermal Rayleigh numbers (Rat = 105), increasing Rab from 10 to 100 causes 4.5% enhancement in average Nusselt number and average Sherwood number decreases by 4.5%. These findings are applicable in various fields of expertise such as ocean ecosystems, oil recovery and fuel cells.
... The trajectories, recorded at a fixed time step, τ ο = 1/30th s given by the video frame rate and, at multiples of τ ο from τ = 2τ ο to τ = 10τ ο . The average of turning angle, μ δ (θ,τ), was calculated as a function of the direction (θ) at each time step (τ) following Roberts (1970), Hill and Häder (1997), Roberts and Deacon (2002), Roberts (2006), Codling et al. (2008). ...
Phytoplankton sinking is an important property that can determine community composition, affecting nutrient and light absorption in the photic zone, and material loss to the deep ocean also influencing biogeochemical cycling. To date, the difficulty in exploring the sinking processes is partly due to methodological limitations in measuring phytoplankton sinking rate. However, in the last decade, works have illustrated various methods based on non-invasive and low perturbing approaches (laser scanner, video-microscopy, fluorescence spectroscopy). In this study, we review the methods for sinking rate estimation and describe the Plankton Tracker, a novel integrated system to investigate in vivo the dynamic sinking behaviour of phytoplankton. Plankton Tracker is composed by a long-distance objective and a video-microscopy facility coupled with a specifically developed image analysis system (Plankton Tracker Elements). Diatoms and dinoflagellates were individually-tracked. By processing recorded videos and images, sinking traits (e.g. trajectory length and complexity, sinking velocity, turning angle) were obtained simultaneously with morphological traits (e.g. linear dimensions). We also applied Biased Correlated Random Walk model to sinking traits. Particularly, we found linear motion in the forming-colony Thalassiosira sp. and sinusoidal reorientation in the dinoflagellate Scrippsiella trochoidea. Our results added additional complexity to diatom suspension of the large Coscinodiscus sp. In this view, Plankton Tracker approach is likely to provide more insights towards the exploration of dynamic sinking behavior in phytoplankton.
... Yet how these microorganisms feel the gravitational field is not fully understood. Several possible origins have been pinpointed: physiological mechanisms [62,157,159]; bottom heaviness [64,61] (inhomogeneity in the swimmer density) or fore-aft asymmetry [160,161,158,63,162]. ...
One may simply be amazed in front of the diversity and complexity of life. Yet, and maybe even more bewildering, living systems all share common hallmarks: compartmentalization, growth and division, information processing, energy transduction and adaptability. In particular, the mobility plays a crucial role in the competitiveness between different species. Physics at microscales is different from the one we are used to at our macroscopic scale. This is why, micro-swimmers have developed specific strategies to induce motion. The understanding of such strategies is crucial at the fundamental level to apprehend the behavior of biological micro-swimmer, but also to achieve artificial locomotion in a surrounding fluid at the micron-scale, in order to perform a multitude of tasks in technical and medical applications (transport, mixing), which has become a central goal of nanoscience. In this context, biological and artificial micro-swimmers have been intensively studied, and we place our study in the framework of swimming in a realistic and complex environment, in the case where external factors (confinement, external force, other swimmers) may influence the swimming properties. In this work, using microfluidic, we create, put into complex situation and observe a model swimmer: a pure water swimming droplet in an outer oil-micelle solution. It was shown that the droplet motion emerges from the nonlinear coupling of hydrodynamics and advection-diffusion of micelles filled with water. We first study the effect of confinement on such geometries using confocal PIV in 3D. The presence of one wall breaks the natural axisymmetry of the flow field. We propose a simplified analytical formulation taking into account the presence of the wall and the effect of buoyancy. This model accounts for the far field hydrodynamic of the droplet close to a wall that differs from the no-wall case. We then look at more confined geometries using glass capillary microfluidic. The velocity of the droplet decreases with increasing confinement; but surprisingly; it saturates at a non-zero value for droplets bigger than the channel height: even very long droplets swim. In more complex geometries, such as stretched capillaries; the droplet elongates while swimming, and amazingly, may undergo successive spontaneous splitting events for high enough confinement. We show that this behavior comes from a saturation in the swollen micelles concentration along the droplet length. External force - such as gravity – also influence the droplet behavior. In 2D, by observing a swimming droplet on an inclined plane, we show that gravity orients the droplet, and that under strong gravity, the droplet’s velocity is more than the simple additivity of the gravity and activity. This is discussed in the light of a theoretical study of the instability mechanism under an external force. The droplet in 1D exhibit a similar behavior, but is also able to swim against gravity. Finally, we investigate their collective dynamics in a 1D micro-fluidic channel. We observe experimentally a rich phenomenology: neighboring droplets align and form large trains. Exanimating the interactions between two "colliding" droplets shows that alignment rises from the interplay between velocity fluctuations and the absence of Galilean invariance. Taking these observations as the basis for a minimalistic 1D model of active particles and combining analytical and numerical arguments, we show that the system exhibits a transition to collective motion. Altogether, the swimming droplet shares numerous similarities with living system: compartmentalization (a droplet), division (under confinement), energy transduction (by thermodynamic relaxation) and adaptability (through the swimming). Beyond the simple understanding of our peculiar system, these studies give insight on various phenomena at the interface of hydrodynamics, physico-chemical engineering and active matter.
... Of particular interest to this research are the mesoscale organisms such as zooplankton; their sizes exceed the Kolmogorov scale but are within the inertial subrange of turbulence. Earlier studies have shown that the orientation of an organism plays a significant role in their gravitaxis, settlement, and directed swimming (Roberts and Deacon, 2002). Besides, the active swimming characteristics of such organisms add further complexity to the problem (Michalec et al., 2017). ...
Turbulence induces rotation in the living and the non-living materials in the ocean. The time scale of rotation for a living organism is important in understanding an organism's feeding efficiency, mating, prey capture rate, etc. This time scale is also crucial for understanding the migration of non-living materials such as microplastics. Herein, we investigate the tumbling motion of mesoscale particles that resemble organisms of intermediate size, such as zooplankton that appear in the ocean. Using time-resolved measurements of the orientation of rigid inertial fibers in a turbulence-tank, we analyze the autocorrelation of their tumbling rate. The correlation time (τd) is well-predicted by Kolmogorov inertial-range scaling based on the fiber length (L) when the fiber inertia can be neglected. For inertial fibers, we propose a simple model considering fiber inertia (measured by a tumbling Stokes number) and a viscous torque which accurately predicts both the correlation time and the variance of the tumbling rate. Our measurements and the theoretical model provide a basic understanding of the rotational response of an intermediately sized organism to the surrounding turbulence in its non-active state.
... Observations of phenomena such as those described above have stimulated an equally large range of theoretical models to see if the observations can be explained by physical processes alone, without requiring an understanding of biological (or chemical) signalling or intracellular processes. Continuum models have been very successful for dilute suspensions, in which the cells interact with their environment but not with each other: bioconvection results from either a gravitational instability when the upswimming of dense cells leads to a gravitationally unstable density profile, or a gyrotactic instability in which the cell's non-uniform density or geometric asymmetry causes them to be reoriented in a shear flow (Childress et al. 1975;Kessler 1986;Pedley & Kessler 1992;Roberts & Deacon 2002). Even when gravity is unimportant the stresses applied by the cell's swimming motions (a swimmer acts as a force dipole or stresslet) lead to instability and random bulk motions (Pedley & Kessler 1990;Simha & Ramaswamy 2002;Saintillan & Shelley 2008). ...
A concentrated, vertical monolayer of identical spherical squirmers, which may be bottom-heavy, and which are subjected to a linear shear flow, is modelled computationally by two different methods: Stokesian dynamics, and a lubrication-theory-based method. Inertia is negligible. The aim is to compute the effective shear viscosity and, where possible, the normal stress differences as functions of the areal fraction of spheres , the squirming parameter (proportional to the ratio of a squirmer's active stresslet to its swimming speed), the ratio Sq of swimming speed to a typical speed of the shear flow, the bottom-heaviness parameter , the angle that the shear flow makes with the horizontal, and two parameters that define the repulsive force that is required computationally to prevent the squirmers from overlapping when their distance apart is less than a critical value , where is very small and a is the sphere radius. The Stokesian dynamics method allows the rheological quantities to be computed for values of up to 0.75; the lubrication-theory method can be used for . A major finding of this work is that, despite very different assumptions, the two methods of computation give overlapping results for viscosity as a function of in the range . This suggests that lubrication theory, based on near-field interactions alone, contains most of the relevant physics, and that taking account of interactions with more distant particles than the nearest is not essential to describe the dominant physics.
... Yet how these microorganisms feel the gravitational field is not fully understood. Several possible origins have been pinpointed: physiological mechanisms [62,157,159]; bottom heaviness [64,61] (inhomogeneity in the swimmer density) or fore-aft asymmetry [160,161,158,63,162]. ...
One may simply be amazed in front of the diversity and complexity of life. Yet, and maybe even more bewildering, living systems all share common hallmarks: compartmentalization, growth and division, information processing, energy transduction and adaptability. In particular, the mobility plays a crucial role in the competitiveness between different species. Physics at microscales is different from the one we are used to at our macroscopic scale. This is why, micro-swimmers have developed specific strategies to induce motion. The understanding of such strategies is crucial at the fundamental level to apprehend the behavior of biological micro-swimmer, but also to achieve artificial locomotion in a surrounding fluid at the micron-scale, in order to perform a multitude of tasks in technical and medical applications (transport, mixing), which has become a central goal of nanoscience. In this context, biological and artificial micro-swimmers have been intensively studied, and we place our study in the framework of swimming in a realistic and complex environment, in the case where external factors (confinement, external force, other swimmers) may influence the swimming properties. In this work, using microfluidic, we create, put into complex situation and observe a model swimmer: a pure water swimming droplet in an outer oil-micelle solution. It was shown that the droplet motion emerges from the nonlinear coupling of hydrodynamics and advection-diffusion of micelles filled with water. We first study the effect of confinement on such geometries using confocal PIV in 3D. The presence of one wall breaks the natural axisymmetry of the flow field. We propose a simplified analytical formulation taking into account the presence of the wall and the effect of buoyancy. This model accounts for the far field hydrodynamic of the droplet close to a wall that differs from the no-wall case. We then look at more confined geometries using glass capillary microfluidic. The velocity of the droplet decreases with increasing confinement; but surprisingly; it saturates at a non-zero value for droplets bigger than the channel height: even very long droplets swim. In more complex geometries, such as stretched capillaries; the droplet elongates while swimming, and amazingly, may undergo successive spontaneous splitting events for high enough confinement. We show that this behavior comes from a saturation in the swollen micelles concentration along the droplet length. External force - such as gravity – also influence the droplet behavior. In 2D, by observing a swimming droplet on an inclined plane, we show that gravity orients the droplet, and that under strong gravity, the droplet’s velocity is more than the simple additivity of the gravity and activity. This is discussed in the light of a theoretical study of the instability mechanism under an external force. The droplet in 1D exhibit a similar behavior, but is also able to swim against gravity. Finally, we investigate their collective dynamics in a 1D micro-fluidic channel. We observe experimentally a rich phenomenology: neighboring droplets align and form large trains. Exanimating the interactions between two "colliding" droplets shows that alignment rises from the interplay between velocity fluctuations and the absence of Galilean invariance. Taking these observations as the basis for a minimalistic 1D model of active particles and combining analytical and numerical arguments, we show that the system exhibits a transition to collective motion. Altogether, the swimming droplet shares numerous similarities with living system: compartmentalization (a droplet), division (under confinement), energy transduction (by thermodynamic relaxation) and adaptability (through the swimming). Beyond the simple understanding of our peculiar system, these studies give insight on various phenomena at the interface of hydrodynamics, physico-chemical engineering and active matter.
... intensively. Many sophisticated physiological methods, such as computer-based movement analysis, inhibitor studies, and electrophysiology were employed [28][29][30][31][32][33][34][35][36]. Despite of a large number of valuable physiological results, the number of proteome or transcriptome-related molecular studies in these organisms is extremely limited. ...
Introduction:
Microgravity (µg) is an extreme stressor for plants, animals and humans and influences biological systems. Humans in space experience various health problems during and after a long-term stay in orbit. Various studies have demonstrated alterations in structural and molecular dynamics within the cellular milieu of plants, bacteria, microorganisms, animals and cells. These data were obtained by proteomics investigations applied in gravitational biology to elucidate changes in the proteome occurring when cells or organisms were exposed to real µg (r-µg) and simulated µg (s-µg). Areas covered: In this review, we summarize the current knowledge about the impact of µg on the proteome in plant, animal and human cells. The literature suggests that µg impacts the proteome and thus various biological processes such as angiogenesis, apoptosis, cell adhesion, cytoskeleton, extracellular matrix proteins, migration, proliferation, stress response, and signal transduction. The changes in cellular function depend on the respective cell type. Expert commentary: This data is important for the topics of gravitational biology, tissue engineering, cancer research and translational regenerative medicine. Moreover, it may provide new ideas for countermeasures to protect the health of future space travelers.
... expressed by the existence of certain cell receptors accepting gravistimulus and, through signaling chains, causing the change in the direction the cell moves (Hemmersbach and Häder, 1999). The physical aspect of gravitaxis is visible in the occurrence of hydrodynamic (Hagen et al., 2014;Roberts and Deacon, 2002) or density gradient (Kessler, 1985) torque which changes the orientation of the cell. The speed of some microorganisms depends on their orientation within the gravity field (excluding buoyancy of the whole cells). ...
A model for overcoming the gravity by sea-picoplankton cells is proposed here. It is based on different means of escaping from potential predators used by cells of co-existing picoplankton species. These different means cause friction anisotropy of motile cells with strong antipredator behavior (AB). According to equations of stochastic movement used in this model for picoplankton cells with strong AB, collocated with high concentration of cells with weak or absent AB, the sedimentation can be considerably overcome.
... Gravitaxis, and hence gyrotaxis (Kessler 1985), is negligible for spermatozoa (Winet et al. 1984) but evident in motile flagellates as well as ciliates (Durham et al. 2009;Roberts and Deacon 2002). Due to a bottom-heavy mass distribution as well as resistive force acting on an asymmetrical geometry (Ten Hagen et al. 2014), these cells tend to be orientated by gravity such that propulsion will lead to a net upward motion, a process known as negative gravitaxis. ...
The validity of some commonly used models and approximations in theoretical studies of microscale locomotion are reviewed, by taking reference from conclusions drawn in experimental and numerical studies. The first model, resistive force theory, neglects interactions and results in vastly varying degrees of accuracy across different studies. The second, slender body theory, uses a series of force singularities to determine the velocity field and generally gives a single-digit percentage error. The method of regularized Stokeslet is the third model, which replaces each force singularity with a blob of distributed forces, and can be as accurate as 99.5 %. In the subsequent part of this review, we look into conditions under which some general assumptions are valid. The studies reviewed show unanimous agreement that wall effects and cell-to-cell interactions can be neglected when the cells are spaced an order of magnitude of their length scale away. In addition, the sedimentation velocity of motile spermatozoa as well as bacteria is concluded to be insignificant. Evidence is also presented to show that Brownian motion of particles larger than 10 µm is negligible. The effect of rheotaxis have to be considered when dealing with shear flow, as the velocity can be modified by over a tenth of a cell’s free-swimming velocity.
... As with other raphidophytes, H. akashiwo has two flagella, and both can be used to provide the cell the ability to change the parameters of their helical swimming path (Fenchel, 2001). Therefore, the variability in movement behaviors among isolates could have resulted from differences in flagella morphology or the cell's physiological capacity for propulsion (Raven and Richardson, 1984;Fenchel, 2001;Roberts and Deacon, 2002). The current findings indicate that cell size cannot be used to estimate movement behaviors nor to predict dispersal and distribution among strains of H. akashiwo. ...
Motility is a key trait that phytoplankton utilize to navigate the heterogeneous marine environment. Quantifying both intra- and inter-specific variability in trait distributions is key to utilizing traits to distinguish groups of organisms and assess their ecological function. Because examinations of intra-specific variability are rare, here we measured three-dimensional movement behaviors and distribution patterns of seven genetically distinct strains of the ichthyotoxic raphidophyte, Heterosigma akashiwo. Strains were collected from different ocean basins but geographic distance between isolates was a poor predictor of genetic relatedness among strains. Observed behaviors were significantly different among all strains examined, with swimming speed and turning rate ranging from 33–115 μm s⁻¹ and 41–110° s⁻¹, respectively. Movement behaviors were consistent over at least 12 h, and in one case identical when measured several years apart. Movement behaviors were not associated with a specific cell size, carbon content, genetic relatedness, or geographic distance. These strain-specific behaviors resulted in algal populations that had distinct vertical distributions in the experimental tank. This study demonstrates that the traits of genetic identity and motility can provide resolution to distinguish strains of species, where variations in size or biomass are insufficient characteristics.
... For example, a flagellum protruding along the major axis renders a cell effectively more elongated than its body alone, as was shown for bacteria [10,26]. Second, the drag associated with flagellar beating also impacts a cell's rotational dynamics in flow [27,28]. For example, the waveforms exhibited during a beat cycle by the two flagella of Chlamydomonas, an alga that swims using a breaststroke motion, reduce the organism's effective aspect ratio when compared with the geometrical aspect ratio of the cell body alone [28]. ...
Fluid flow, ubiquitous in natural and man-made environments, has the potential to profoundly impact the transport of microorganisms, including phytoplankton in aquatic habitats and bioreactors. Yet, the effect of ambient flow on the swimming behaviour of phytoplankton has remained poorly understood, largely owing to the difficulty of observing cell-flow interactions at the microscale. Here, we present microfluidic experiments where we tracked individual cells for four species of motile phytoplankton exposed to a spatially non-uniform fluid shear rate, characteristic of many flows in natural and artificial environments. We observed that medium-to-high mean shear rates (1-25 s(-1)) produce heterogeneous cell concentrations in the form of regions of accumulation and regions of depletion. The location of these regions relative to the flow depends on the cells' propulsion mechanism, body shape and flagellar arrangement, as captured by an effective aspect ratio. Species having a large effective aspect ratio accumulated in the high-shear regions, owing to shear-induced alignment of the swimming orientation with the fluid streamlines. Species having an effective aspect ratio close to unity exhibited little preferential accumulation at low-to-moderate flowrates, but strongly accumulated in the low-shear regions under high flow conditions, potentially owing to an active, behavioural response of cells to shear. These observations demonstrate that ambient fluid flow can strongly affect the motility and spatial distribution of phytoplankton and highlight the rich dynamics emerging from the interaction between motility, morphology and flow.
... He developed a general mathematical theory of geotaxis to desc:::* the motion of these organisms under gravity, and the predictions of the ::qory were compared with measurements in suspensions of Parameci,a in lo-E vertical columns (Roberts 1970). In a later study, examined in more details the movement of gravitactic ciliates such as Pan;-'*,ctum (Roberts and Deacon 2002). They found that the shape-dependent '-"::entation plays an important role in the gravitactic responses of Parameci':-,.. ...
... This phenomenon is not observed in non-swimming Paramecium [80]. Conversely, there also exist passive responses for which a physical field, such as magnetic field, may directly orient an organism without eliciting a physiological change [94,104,33]. ...
Intense static magnetic fields of up to 31 Tesla were used as a novel tool to manipulate the swimming mechanics of unicellular organisms. It is shown that homogenous magnetic fields alter the swimming trajectories of the single cell protozoan Paramecium caudatum, by aligning them parallel to the applied field. Immobile neutrally buoyant paramecia also oriented in magnetic fields with similar rates as the motile ones. It was established that the magneto-orientation is mostly due to the magnetic torques acting on rigid structures in the cell body and therefore the response is a non-biological, passive response. From the orientation rate of paramecia in various magnetic field strengths, the average anisotropy of the diamagnetic susceptibility of the cell was estimated. It has also been demonstrated that magnetic forces can be used to create increased, decreased and even inverted simulated gravity environments for the investigation of the gravi-responses of single cells. Since the mechanisms by which Earth's gravity affects cell functioning are still not fully understood, a number of methods to simulate different strength gravity environments, such as centrifugation, have been employed. Exploiting the ability to exert magnetic forces on weakly diamagnetic constituents of the cells, we were able to vary the gravity from -8 g to 10 g, where g is Earth's gravity. Investigations of the swimming response of paramecia in these simulated gravities revealed that they actively regulate their swimming speed to oppose the external force. This result is in agreement with centrifugation experiments, confirming the credibility of the technique. Moreover, the Paramecium's swimming ceased in simulated gravity of 10 g, indicating a maximum possible propulsion force of 0.7 nN. The magnetic force technique to simulate gravity is the only earthbound technique that can create increased and decreased simulated gravities in the same experimental setup. These findings establish a general technique for applying continuously variable forces to cells or cell populations suitable for exploring their force transduction mechanisms.
... According to a present model, gravitaxis is most likely triggered by mechano-sensitive membrane channels followed by a series of subsequent physiological mechanism, among others membrane potential changes, increased calcium and cAMP concentration, involvement of calmodulin ( Häder 1999a, Streb et al. 2002 ). An additional physical alignment component caused by body asymmetry is possible (Roberts and Deacon 2002, Richter et al. 2002c). In earlier experiments performed with fluorescent calcium indicators increased intracellular calcium was observed upon cell stress (drought, high light exposure). ...
Euglena gracilis responds to abiotic stress factors (high light, salinity, heavy metals) with a sign change of its gravitactic behavior. This phenomenon is oxygen dependent and can be suppressed by the application of the reductant dithionite. It is not mediated by the photoreceptor since also blind mutants change their movement behavior upon high light exposure. It is also not mediated by the chloroplasts since the gravitactic sign change was also found in white, chloroplast-free mutants. The NO radical donor SNAP and the NO cleaver carboxy-PTIO had no obvious effects on gravitaxis or gravitactic sign change, respectively, indicating that NO radicals are not likely involved in gravitactic sign change. Gravitactic sign change was suppressed when oxygen was removed by flushing the cell suspension with nitrogen. Also, the addition of the radical scavengers Trolox, ascorbic acid or potassium cyanide abolished or reduced gravitactic sign change. Quantification of reactive oxygen species (ROS) in the cells indicated that these treatments reduced the evolution of ROS. Furthermore, addition of hydrogen peroxide induced gravitactic sign change in the absence of external stress factors. These results indicate that gravitactic sign change is triggered by ROS (most likely hydrogen peroxide) which are probably produced by cytochrome-c-oxidase in the mitochondria. The clear responses of Euglena to abiotic stress factors suggest that these cells are probably interesting model systems in the study of stress signaling.
... From a different barbellFigure 7: The asymmetrical barbell model consists of two spheres of unequal radii and equal density. model, originally proposed by Roberts[9], and then further developed[10], we can model reorientation arising from front-rear asymmetries.Fig.7 shows a barbell consisting of two spheres of equal density ...
... Other behaviors leading to a similar reorientation include swimming patterns ͑due to flagella 16 or whole body swimming 17 ͒, active reorientation using a gravitational sensor, 18 or fore-aft asymmetry. 19 In flow viscous forces exerted by the fluid on the cell interact with gravitaxis causing a modification of the cell orientation and hence its mean swimming velocity. The interaction between the viscous and reorienting torques is termed gyrotaxis. ...
In this paper we consider spherical organisms whose orientations are determined by the combination of viscous torque with a self-righting torque either due to bottom-heaviness, shape, or through preferential swimming due to active sensing: a phenomenon known as gyrotaxis. We restate the stable equilibrium orientation of such an organism in a homogeneous shear flow, which exists for all shear values if the two torques are not parallel, and only when the nondimensional shear is below a critical value if the torques are parallel. We calculate the long-term trajectories of such an organism in a flow, which consists of linear motion in the direction of the vorticity and a conic section in the plane of shear, giving, for instance, helical motion in the case of pure rotation. We compute the orientation probability density function when the reorientation is coupled with rotational diffusion and show that including rotational diffusion reduces the mean swimming speed and rotates the mean equilibrium orientation toward the vertical. Numerical simulation of stochastic gyrotactic cells in synthetic turbulence demonstrates the interaction between swimming and more general flow fields. The calculation of the mean drift and diffusion demonstrates that the expected transition between swimming-dominated transport for low kinetic energy dissipation rate and turbulence-dominated transport for large occurs over dissipation rates such motile organisms experience in nature 10 −10 –10 −3 m 2 s −3 . © 2010 American Institute of Physics.
The fluid dynamics of microswimmers has received attention from the fields of microbiology, microrobotics, and active matter. Microorganisms have evolved organelles termed cilia for propulsion through liquids. Each cilium periodically performs effective and recovery strokes, creating a metachronal wave as a whole and developing a propulsive force. One well-established mathematical model of ciliary swimming is the squirmer model, which focuses on surface squirming velocities. This model is also useful when studying active colloids and droplets. The squirmer model has been recently used to investigate the behaviors of microswimmers in complex environments, their collective dynamics, and the characteristics of active fluids. Efforts have also been made to broaden the range of applications beyond the assortment permitted by the squirmer model, which was established to specifically represent ciliary flow and incorporate biological features. The stress swimmer model imposes stresses above the cell body surface that enforce the no-slip condition. The ciliated swimmer model precisely reproduces the behaviors of each cilium that engages in mutual hydrodynamic interactions. Mathematical models have improved our understanding of various microbial phenomena, including cell–cell and cell–wall interactions and energetics. Here, I review recent advances in the hydrodynamics of ciliary swimming and then discuss future challenges.
Expected final online publication date for the Annual Review of Fluid Mechanics, Volume 56 is January 2024. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
Diel vertical migration (DVM), the diurnal exodus of motile phytoplankton between the light- and nutrient-rich aquatic regions, is governed by endogenous biological clocks. Many species exhibit irregular DVM patterns wherein out-of-phase gravitactic swimming–relative to that expected due to the endogenous rhythm–is observed. How cells achieve and control this irregular swimming behavior remains poorly understood. Combining local environmental monitoring with behavioral and physiological analyses of motile bloom-forming Heterosigma akashiwo cells, we report that phytoplankton species modulate their DVM pattern by progressively tuning local pH, yielding physiologically equivalent yet behaviorally distinct gravitactic sub-populations which remain separated vertically within a visibly homogeneous cell distribution. Individual and population-scale tracking of the isolated top and bottom sub-populations revealed similar gravitactic (swimming speed and stability) and physiological traits (growth rate and maximum photosynthetic yield), suggesting that the sub-populations emerge due to mutual co-existence. Exposing the top (bottom) sub-population to the spent media of the bottom (top) counterpart recreates the emergent vertical distribution, while no such phenomenon was observed when the sub-populations were exposed to their own spent media. A model of swimming mechanics based on the quantitative analysis of cell morphologies confirms that the emergent sub-populations represent distinct swimming stabilities, resulting from morphological transformations after the cells are exposed to the spent media. Together with the corresponding night-time dataset, we present an integrated picture of the circadian swimming, wherein active chemo-regulation of the local environment underpins motility variations for potential ecological advantages via intraspecific division of labor over the day-night cycle. This chemo-regulated migratory trait offers mechanistic insights into the irregular diel migration, relevant particularly for modelling phytoplankton transport, fitness and adaptation as globally, ocean waters see a persistent drop in the mean pH.
One sentence summary
Active regulation of local pH diversifies the diel vertical migration of motile phytoplankton.
Gravity has an unavoidable effect on all living organisms inhabiting fluidic surroundings. To investigate the spatial distribution of bacteria in quiescent fluids and their rheotactic behavior in shear flows under buoyancy, we adjust the buoyant force to regulate bacterial swimming in a microfluidic channel. It is found that swimming bacteria of Escherichia coli exhibit an obvious vertical separation when exposed to a medium with high density and gradually gather close to the up wall within minutes. The bacterial population presents a net upward number flux, which enhances the trapping of motile bacteria onto the up surface as a result of buoyancy force apart from the hydrodynamic and kinematic interactions in quiescent fluids. When flow is imposed into the channel, the buoyancy effect is however significantly suppressed. Additionally, the drift velocity perpendicular to the buoyancy vector as a result of chirality-induced transverse swimming decreases with buoyancy force. However, this transverse drift capability is recovered after excluding the intrinsic swimming motility in a quiescent medium. Failing to escape from the trapping as a result of buoyant force allows for a facile separation of bacteria along the vertical direction. The findings also offer a controllable way to redisperse and homogenize the bacteria distribution close to walls by imposing a weak shear flow.
A small spheroid settling in a quiescent fluid experiences an inertial torque that aligns it so that it settles with its broad side first. Here we show that an active particle experiences such a torque too, as it settles in a fluid at rest. For a spherical squirmer, the torque is where is the swimming velocity, is the settling velocity in the Stokes approximation and is the equivalent fluid mass. This torque aligns the swimming direction against gravity: swimming up is stable, swimming down is unstable.
Nutrient availability, along with light and temperature, drives marine primary production. The ability to migrate vertically, a critical trait of motile phytoplankton, allows species to optimize nutrient uptake, storage, and growth. However, this traditional view discounts the possibility that migration in nutrient-limited waters may be actively modulated by the emergence of energy-storing organelles. Here, we report that bloom-forming raphidophytes harness energy-storing cytoplasmic lipid droplets (LDs) to biomechanically regulate vertical migration in nutrient-limited settings. LDs grow and translocate directionally within the cytoplasm, steering strain-specific shifts in the speed, trajectory, and stability of swimming cells. Nutrient reincorporation restores their swimming traits, mediated by an active reconfiguration of LD size and coordinates. A mathematical model of cell mechanics establishes the mechanistic coupling between intracellular changes and emergent migratory behavior. Amenable to the associated photophysiology, LD-governed behavioral shift highlights an exquisite microbial strategy toward niche expansion and resource optimization in nutrient-limited oceans.
As open oceans continue to warm, modified currents and enhanced stratification exacerbate nitrogen and phosphorus limitation, constraining primary production. The ability to migrate vertically bestows motile phytoplankton a crucial–albeit energetically expensive–advantage toward vertically redistributing for optimal growth, uptake and resource storage in nutrient-limited water columns. However, this traditional view discounts the possibility that the phytoplankton migration strategy may be actively selected by the storage dynamics when nutrients turn limiting. Here we report that storage and migration in phytoplankton are coupled traits, whereby motile species harness energy storing lipid droplets (LDs) to biomechanically regulate migration in nutrient limited settings. LDs grow and translocate–directionally–within the cytoplasm to accumulate below the cell nucleus, tuning the speed, trajectory and stability of swimming cells. Nutrient reincorporation reverses the LD translocation, restoring the homeostatic migratory traits measured in population-scale millifluidic experiments. Combining intracellular LD tracking and quantitative morphological analysis of red-tide forming alga, Heterosigma akashiwo , along with a model of cell mechanics, we discover that the size and spatial localization of growing LDs govern the ballisticity and orientational stability of migration. The strain-specific shifts in migration which we identify here are amenable to a selective emergence of mixotrophy in nutrient-limited phytoplankton. We rationalize these distinct behavioral acclimatization in an ecological context, relying on concomitant tracking of the photophysiology and reactive oxygen species (ROS) levels, and propose a dissipative energy budget for motile phytoplankton alleviating nutrient limitation. The emergent resource acquisition strategies, enabled by distinct strain-specific migratory acclimatizing mechanisms, highlight the active role of the reconfigurable cytoplasmic LDs in guiding vertical movement. By uncovering the mechanistic coupling between dynamics of intracellular changes to physiologically-governed migration strategies, this work offers a tractable framework to delineate diverse strategies which phytoplankton may harness to maximize fitness and resource pool in nutrient-limited open oceans of the future.
One sentence summary
Phytoplankton harness reconfigurable lipid droplets to biomechanically tune migratory strategies in dynamic nutrient landscapes.
Experiments on autophoretic bimetallic nanorods propelling within a fuel of hydrogen peroxide show that tail-heavy swimmers preferentially orient upwards and ascend along inclined planes. We show that such gravitaxis is strongly facilitated by interactions with solid boundaries, allowing even ultraheavy microswimmers to climb nearly vertical surfaces. Theory and simulations show that the buoyancy or gravitational torque that tends to align the rods is reinforced by a fore-aft drag asymmetry induced by hydrodynamic interactions with the wall.
Highlights • We review the methodological issues in measuring phytoplankton sinking rate • We describe the Plankton Tracker, a novel system to investigate sinking behavior • Diatoms and dinoflagellates were individually-tracked by recording video and images • Sinking and morphological traits were obtained simultaneously • We apply Biased Correlated Random Walk model to sinking traits ABSTRACT Phytoplankton sinking is an important property that can determine community composition, affecting nutrient and light absorption in the photic zone, and influencing biogeochemical cycling via material loss to the deep ocean. To date, the difficulty in exploring the sinking processes is partly due to methodological limitations in measuring phytoplankton sinking rate. However, in the last decade, works have illustrated various methods based on non-invasive and low perturbing approaches (laser scanner, video-microscopy, fluorescence spectroscopy). In this study, we review the methods for sinking rate estimation and describe the Plankton Tracker, a novel integrated system to investigate in
The term “bioconvection” describes hydrodynamic instabilities and patterns in suspensions of biased swimming microorganisms. Hydrodynamic instabilities arise from coupling between cell swimming behaviors; physical properties of the cells, such as density; and fluid flows. For instance, a combination of viscous and gravitational torques can lead to cells swimming toward downwelling fluid. If the cells are more dense than the fluid, then a gyrotactic instability results. Phototaxis describes the directed response of cells to light, which can also lead to instability. Bioconvection represents a classic system where macroscopic phenomena arise from microscopic cellular behavior in relatively dilute systems. There are ecological consequences for bioconvection and the mechanisms involved as well as potential for industrial exploitation. The focus of this review is on progress measuring and modeling gyrotactic and phototactic bioconvection. It builds on two earlier reviews of bioconvection and recent interest in active matter, describing progress and highlighting open problems.
The recent advances in the micro/nanomotor field have shown great progress in the propulsion of such devices by fuel-free mechanisms. Light, as an abundant and natural source, has been demonstrated to be a promising external field to wirelessly induce the motion of these tiny micro/nanomachines, without the need of any toxic fuel or complex system set-up. This tutorial review covers the most representative examples of light-driven micro/nanomotors developed so far, which self-propelled exclusively under fuel-free conditions. Their different swimming behaviors triggered by light stimuli, divided into four main categories (schooling, phototaxis, gravitaxis and directional motion), are discussed along with their similarities with the motion modes of microorganisms. Moreover, the main parameters that influence the motion of light-driven photocatalytic-based micro/nanomotors as well as alternative strategies to develop more efficient systems are also discussed.
For natural microswimmers, the interplay of swimming activity and external flow can promote robust motion, e.g. propulsion against ("upstream rheotaxis") or perpendicular to the direction of flow. These effects are generally attributed to their complex body shapes and flagellar beat patterns. Here, using catalytic Janus particles as a model experimental system, we report on a strong directional response that occurs for spherical active particles in a channel flow. The particles align their propulsion axes to be nearly perpendicular to both the direction of flow and the normal vector of a nearby bounding surface. We develop a deterministic theoretical model of spherical microswimmers near a planar wall that captures the experimental observations. We show how the directional response emerges from the interplay of shear flow and near-surface swimming activity. Finally, adding the effect of thermal noise, we obtain probability distributions for the swimmer orientation that semi-quantitatively agree with the experimental distributions.
Marine phytoplankton inhabit a dynamic environment where turbulence, together with nutrient and light availability, shapes species fitness, succession and selection. Many species of phytoplankton are motile and undertake diel vertical migrations to gain access to nutrient-rich deeper layers at night and well-lit surface waters during the day. Disruption of this migratory strategy by turbulence is considered to be an important cause of the succession between motile and non-motile species when conditions turn turbulent. However, this classical view neglects the possibility that motile species may actively respond to turbulent cues to avoid layers of strong turbulence. Here we report that phytoplankton, including raphidophytes and dinoflagellates, can actively diversify their migratory strategy in response to hydrodynamic cues characteristic of overturning by Kolmogorov-scale eddies. Upon experiencing repeated overturning with timescales and statistics representative of ocean turbulence, an upward-swimming population rapidly (5-60 min) splits into two subpopulations, one swimming upward and one swimming downward. Quantitative morphological analysis of the harmful-algal-bloom-forming raphidophyte Heterosigma akashiwo together with a model of cell mechanics revealed that this behaviour was accompanied by a modulation of the cells' fore-aft asymmetry. The minute magnitude of the required modulation, sufficient to invert the preferential swimming direction of the cells, highlights the advanced level of control that phytoplankton can exert on their migratory behaviour. Together with observations of enhanced cellular stress after overturning and the typically deleterious effects of strong turbulence on motile phytoplankton, these results point to an active adaptation of H. akashiwo to increase the chance of evading turbulent layers by diversifying the direction of migration within the population, in a manner suggestive of evolutionary bet-hedging. This migratory behaviour relaxes the boundaries between the fluid dynamic niches of motile and non-motile phytoplankton, and highlights that rapid responses to hydrodynamic cues are important survival strategies for phytoplankton in the ocean.
Recent studies show that spherical motile microorganisms in turbulence subject to gravitational torques gather in down-welling regions of the turbulent flow. By analyzing a statistical model we analytically compute how shape affects the dynamics, preferential sampling, and small-scale spatial clustering. We find that oblong organisms may spend more time in up-welling regions of the flow, and that all organisms are biased to regions of positive fluid-velocity gradients in the upward direction. We analyze small-scale spatial clustering and find that oblong particles may either cluster more or less than spherical ones, depending on the strength of the gravitational torques.
The motion of particles suspended in environmental turbulence is relevant to
many scientific fields, from sediment transport to biological interactions to
underwater robotics. At very small scales and simple shapes, we are able to
completely mathematically describe the motion of inertial particles; however,
the motion of large aspherical particles is significantly more complex, and
current computational models are inadequate for large or highly-resolved
domains. Therefore, we seek to experimentally investigate the coupling between
freely suspended particles and ambient turbulence. A better understanding of
this coupling will inform not only engineering and physics, but the
interactions between small aquatic organisms and their environments. We explore
the roles of shape and buoyancy on the motion of passive particles in
turbulence, and allow these particles to serve as models for meso-scale aquatic
organisms. The results of this study will allow us to more accurately predict
the motion of aspherical particles, giving new insights into oceanic carbon
cycling, industrial processes, and other important topics. This analysis will
also shed light onto biological questions of navigation, reproduction, and
predator-prey interaction by quantifying the turbulence-driven behavior of
meso-scale aquatic organisms, allowing researchers to sift out passive vs.
active effects in a behaving organism. Lastly, processes that are directly
dependent on particle dynamics (e.g., sediment transport, industrial processes)
will be informed by our results.
The diversity of the morphologies, propulsion mechanisms, flow environments, and behaviors of planktonic microorganisms has long provided inspiration for fluid physicists, with further intrigue provided by the counterintuitive hydrodynamics of their viscous world. Motivation for studying the fluid dynamics of microplankton abounds, as microorganisms support the food web and control the biogeochemistry of most aquatic environments, particularly the oceans. In this review, we discuss the fluid physics governing the locomotion and feeding of individual planktonic microorganisms (⩽1mm). In the past few years, the field has witnessed an increasing number of exciting discoveries, from the visualization of the flow field around individual swimmers to linkages between microhydrodynamic processes and ecosystem dynamics. In other areas, chiefly the ability of microorganisms to take up nutrients and sense hydromechanical signals, our understanding will benefit from reinvigorated interest, and ample opportunities for...
Uniform suspensions of bottom-heavy, upswimming (gyrotactic) micro-organisms that are denser than water are unstable, through a gravitational mechanism first described by Pedley, Hill & Kessler (J. Fluid Mech., vol. 195, 1988, p. 223). Suspensions of downswimming, head-heavy cells do not experience this instability. In the absence of gravity, a uniform suspension of swimming micro-organisms may be unstable because of the ‘particle stresses’ generated by the swimming cells themselves, each of which acts as a force-dipole or stresslet (Simha & Ramaswamy, Phys. Rev. Lett., vol. 89, 2002, p. 058101). The stresslet strength S is positive for ‘pullers’ such as algae and negative for ‘pushers’ such as bacteria or spermatozoa. In this paper, the combined problem is investigated, with attention being paid also to the effect of rotational diffusivity and to whether the probability density function f(e) for the cells' swimming direction e can be approximated as quasi-steady in calculations of the mean swimming direction, which arises in the cell conservation equation, and the particle stress tensor, which appears in the momentum equation. The existence of both the previous instabilities is confirmed at long wavelength. However, the non-quasi-steadiness of the orientation distribution means that the particle-stress-driven instability no longer arises for arbitrarily small |S|, in the Stokes limit, but requires that the dimensionless stresslet strength (proportional to the product of S and the basic state cell volume fraction no) exceed a critical value involving both viscosity and rotational diffusivity. In addition, a new mode of gravitational instability is found for ‘head-heavy’ cells, even when they exert no particle stresses (S = 0), in the form of weakly growing waves. This is a consequence of unsteadiness in the mean swimming direction, together with non-zero fluid inertia. For realistic parameter values, however, viscosity is expected to suppress this instability.
Geotaxis is demonstrated in the ciliated protozoon Loxodes. This behaviour is mediated by a mechanoreceptor which is probably the Müller body, an organelle characteristic of loxodid ciliates. The geotactic response is sensitive to dissolved oxygen tension: in anoxia or at very low O2 tensions the ciliates tend to swim up and at higher O2 tensions they tend to swim down. This behaviour, in conjunction with a kinetic response allows the ciliates to orientate themselves in vertical O2 gradients and to congregate in their optimum environment. In two appendices, models of the behaviour predicting vertical distribution patterns and considerations of the minimum size of a functional statocyst are offered.
Summary The swimming behaviour of Paramecium was analyzed under natural and experimental hypergravity conditions. Paramecium that swam upwards (in the opposite direction to the gravitational force) along a straight path (straight swimmers) swam more slowly than those swimming downwards. This dependence of the swimming velocity on its direction relative to gravity can be partly interpreted as the consequence of sinking due to gravity if the propulsive force does not vary. The effect was different for Paramecium swimming along a circular path (curved swimmers). The difference in velocity between those swimming upwards and those swimming downwards was substantially smaller than would have been expected from sinking effects with invariant propulsion even after correcting for maximal hydrodynamic wall effects, indicating that Paramecium compensate for sinking caused by gravity by controlling their propulsion. The propulsive velocity evaluated by vector calculus increased both as Paramecium swam more sharply upwards and as the experimental gravitational force was increased. The dependence of propulsion on the swimming direction and on gravity was reduced in a high-density medium of nearly neutral buoyancy, suggesting that the site of gravireception is unlikely to be in the interior of the cell. The differences between straight and curved swimmers are discussed in terms of rapid adaptation of gravireceptors in the cell membrane, desensitization of mechanosensory channels and hyperactivation of ciliary activity in straight swimmers.
Tetrahymena pyriformis is a small protozoan cell, which has less than 10% of the volume of the well-known Paramecium caudatum. We investigated the graviresponses of Tetrahymena in a search of the lower limits of gravitransduction in ciliates. Equilibrated populations of free swimming cells were enclosed in chambers of 2 mm depth testing the orientational and speed responses with the chambers in horizontal or vertical position. For determinations of gravikinesis, the sedimentation rates of cells immobilized by application of two different procedures were measured. Negative gravitaxis was pronounced after turning the chambers from horizontal to vertical position: it settled, after 1 min, toward an orientation coefficient of 0.2. Gravikinesis did not only neutralize the sedimentation rate (= 22 μms-1) but even exceeded that rate by at least 30%. Tetrahymena is thereby the first cell, in which overcompensation of the sedimentation rate was documented. Biophysical considerations suggest a high gravisensitivity of Tetrahymena with channel gating energy being less than 33 times above the thermic noise level.
Sedimentation is part of the swimming behaviour of cells in the gravity field. Determinations of sedimentation are indispensable for ute isolation of graviresponses of cells (gravitaxis, gravikinesis). Using a low-speed microscope centrifuge (NIZEMI) we have determined the sedimentation rates of various protozoan cells (Paramecium tetraurelia, Paramecium caudatum, Tetrahymena pyriformis. Didinium nasutum, and Loxodes striatus) as a function of acceleration up to 7 g, iron uptake, deciliation, two different immobilization agents (NiCl2, MnCl2), and Ca2+ -Mg2+ composition of the experimental solution. The experimental results suggest a linear relationship between sedimentation rate and acceleration up to 2g intercepting the origin (zero sedimentation rate at 0 g). Beyond 2 g the sedimentation curves tend to depart from linearity, depending on cell shape and size, state of feeding, immobilization procedure, and time passed since the onset of sedimentation. The time-dependence of observed mean values of sedimentation rate rises with acceleration due to the continuing loss of the faster sedimenting cells from the observed field. The original distribution of sedimentation velocities of a cell sample can be calculated from the data, as long as cells of the highest velocity classes have not escaped registration. Correcting experimental data for time-dependent modification of the velocity distribution modifies the sedimentation curves toward linearity. A strict linearity of the velocity-g relationship was nevertheless not obtained for g-values beyond 2 g suggesting that the departure from linearity is not only a time-dependent artifact but is affected by additional factors such as the surface properties of cells and the proximity of solid walls enclosing the experimental fluid space.
A flagellated, bottom-heavy micro-organism's swimming direction in a shear flow is determined from a balance between the gravitational and viscous torques (gyrotaxis). Hitherto, the cell has been assumed to be a spheroid and the flagella have been neglected. Here we use resistive-force theory to calculate both the magnitude and the direction of a biflagellate cell's swimming velocity and angular velocity relative to the fluid when there is an arbitrary linear flow far from the cell. We present an idealized model for the flagellar beat but, in calculating the velocity of the fluid relative to an element of a flagellum, the presence of the cell body is not neglected. Results are given for the case of a spherical cell body whose flagella beat in a vertical plane, when the ambient linear flow is in the same vertical plane. Results show that resistive-force theory can be used for organisms where the cell body has significant effect on the flow past the flagella and that the viscous torque on the flagella is a significant term in the torque balance equations. A model is presented for the calculation of a cell's velocity and angular velocity in a shear flow which is valid up to high magnitudes of rate of strain or vorticity. The main application of the results will be to modify a recent continuum model for suspensions of gyrotactic micro-organisms (Pedley & Kessler 1990).
This paper shows how to calculate local equilibrium orientations of inhomogeneous spheroidal particles placed in a flow field. The results can be applied either to dilute suspensions of inert particles or to swimming microorganisms; illustrative examples are chosen with the latter application in mind. The centre of mass of a particle is displaced from the geometric centre C along the axis of symmetry, and the orientation of this axis (represented by the unit vector p) is determined from the balance between the gravitational couple, non-zero when p is not vertical, and the viscous couple exerted by the surrounding fluid. Fluid and particle inertia are neglected. `Local equilibrium' means that p is stationary in a suitable frame of reference, which may be the laboratory frame or one rotating rigidly relative to it, at the values of fluid velocity, vorticity and rate of strain evaluated at C in the absence of the particle. It is also shown how to determine the stability of local equilibria. Stable equilibrium values of p are calculated explicitly for a number of experimentally realizable flow fields, including vertical Poiseuille flow in a pipe, conical sink flow, two-dimensional straining and shearing flows in a vertical plane, and the wake of a falling sphere. The analysis is particularly simple for spherical particles, when the local rate of strain does not contribute to the viscous couple. The results have implications for laboratory manipulation of the trajectories of swimming algae, and for the development of collective behaviour and the existence of critical phenomena in suspensions of them.
The movement of any short length of the tail of a spermatozoon of Psammechinus miliaris and the characteristic changes which it undergoes during each cycle of its displacement through the water can be described in terms of the form and speed of propagation of the bending waves which travel along the tail (Gray, 1953, 1955); the form of the wave depends on the maximum extent of bending reached by each element and on the difference in phase between adjacent elements. The object of this paper is to consider the forces exerted on the tail as it moves relative to the surrounding medium and to relate the propulsive speed of the whole spermatozoon to the form and speed of propagation of the bending waves generated by the tail. The mathematical theory of the propulsive properties of thin undulating filaments has recently been considered by Taylor (1951, 1952) and by Hancock (1953); the present study utilizes and extends their findings but approaches the problem from a somewhat different angle. resistance, and consequently the transverse displacement (Vy) elicits reactions tangential and normal to the surface of the element. The latter force (δNy) has a component(δNysinθ) acting forward along the axis (xx ′) of propulsion; it is this component which counteracts the retarding effect of all the forces acting tangentially to the surface.
A model for collective movement and pattern formation in layered suspensions of negatively geotactic micro-organisms is presented. The motility of the organism is described by an average upward swimming speed U and a diffusivity tensor D. It is shown that the equilibrium suspension is unstable to infinitesimal perturbations when either the layer depth or the mean concentration of the organisms exceeds a critical value. For deep layers the maximum growth rate determines a preferred pattern size explicitly in terms of U and D. The results are compared with observations of patterns formed by the ciliated protozoan Tetrahymena
pyriformis.
1.
We have investigated a physiological component of the gravitaxis of Paramecium using established mechanisms of ciliate mechanosensitivity. The horizontal, up and down swimming rates of cells, and the sedimentation of immobilized specimens were determined. Weak DC voltage gradients were applied to predetermine the Paramecium swimming direction.
2.
An observed steady swimming rate is the vector sum of active propulsion (P), a possible gravity-dependent change in swimming rate (), and rate of sedimentation (S). We approximated P from horizontal swimming. S was measured after cell immobilization.
3.
Theory predicts that the difference between the down and up swimming rates, divided by two, equals the sum of S and . is supposed to be the arithmetic mean of two subcomponents,
a and
p, from gravistimulation of the anterior and posterior cell ends, respectively.
4.
A negative value of (0.038 mm/s) was isolated with
a(0.070 mm/s) subtracting from downward swimming, and
p(0.005 mm/s) adding to upward propulsion. The data agree with one out of three possible ways of gravisensory transduction: outward deformation of the mechanically sensitive lower soma membrane. We call the response a negative gravikinesis because both
a and
p antagonize sedimentation.
Effects of the density of the external medium on gravireception in Loxodes striatus were investigated using Percoll solutions. With increasing density, the swimming rates changed from prevailing in the downward
direction to prevailing in the upward direction. A cellular density of 1.036 g cm−3 was determined measuring direction and speed of sedimenting immobilized cells at different accelerations and medium densities.
Viscosity increases by Percoll were measured and taken into account. At 30% air saturation Loxodes maintained a negative gravikinesis of approximately −27 μm s−1 at external densities corresponding to cellular density (±0.02 g cm−3). Negative gravikinesis decreased gradually to −9 μm s−1 with the density difference rising from 0.020 to 0.036 g cm−3 (=normal). The data indicate the existence of central gravireception, presumably by the Müller organelle, to generate in swimming
Loxodes a constant value of gravikinesis and a bimodal gravitaxis. Peripheral gravireception occurs, in addition to central gravireception,
when the transmembrane density difference exceeds 0.02 g cm−3. Peripheral gravireception can neutralize, in part, gravikinesis as raised by the central gravireceptor. We hypothesize that
both central and peripheral gravireception of Loxodes guide vertical locomotion in gliding and swimming cells.
Two-dimensional swimming tracks of paramecia on the vertical plane were photographed by 4-sec exposure under a dark field. The orientation rate (meaning the degree of upward curvature of the swimming path) was measured under various experimental conditions. The rate decreased in a K-rich medium as well as in a viscous methylcellulose solution. The decreasing tendency of the orientation rate was closely coupled with that of swimming velocity. The rate was also decreased in a solution with a high specific gravity which was made up by heavy water. The specimen did not show geotactic orientation either during or just after the application of an electric current. The motile behavior of the individual specimen under gravity, which was recorded on videotape, showed that a larger angular change in the cell axis was always coupled with an upswing phase of the gyration. The theoretical equations of motion for physical models were set up, and the most probable mechanism was discussed on the basis of the present findings.
The swimming direction of algal cells can be guided so that the cells are focused into a concentrated beam. This directed locomotion, or taxis, results from the orientation of the cells' axes by compensating gravitational and viscous torques. It is named gyrotaxis because of this origin. Gyrotaxis includes rheotaxis1, which is concerned with orientation and locomotion of elongated microorganisms, especially spermatozoa, in fluids with a velocity gradient. I present here a simplified theory of gyrotaxis, together with experimental evidence. A geometrical arrangement is shown whereby the effect can be made the basis of a new method for concentrating and separating motile cells. Unlike standard concentration/separation techniques, the gyrotactic method requires active participation of the cells and can, in principle, distinguish among them on the basis of morphology and swimming behaviour. I also discuss the role of gyrotaxis in the maintenance of naturally occurring descending streams of cells and in bioconvection patterns.
This paper presents evidence that the negative geotactic behavior of Paramecium caudatum takes place by the mechanism of buoyancy-oriented upward swimming. Photographs of swimming pathways of the organisms were completely described by two dynamic equations for the translational motion of the center of gravity of the organism's body and for the rotational motion of the organism's body about its center of gravity, where the rotational torque is induced by a slight difference in position between the center of gravity and the center of buoyancy. It now seems unlikely that complicated mechanisms such as the statocyst mechanism and the gravity-propulsion mechanism, which have been proposed by many investigators, need be considered for other protozoa since preliminary observation and analysis of other ciliates such as Paramecium multimicronucleatum, Paramecium tetraurelia, and Tetrahymena pyriformis also strongly suggested that their negative geotaxis is due to buoyancy-oriented upward swimming.
In this paper it is shown that the variable-density orientation cannot completely account for geotaxis in ciliates such as Paramecium, and it is suggested that the principal cause is a hydrodynamic interaction between the organisms and the medium, the magnitude of which is determined by the size and shape of the organism. Many of the motile protozoa, for example, are characteristically wider at the rear than at the front; because larger objects tend to fall more rapidly in a viscous fluid, the rear tends to sink below the front, thus producing an upward orientation. The nature of the hydro- dynamic interaction is investigated using small-scale models falling through glycerol, and the results are compared with similar experiments on immobilized paramecia. A general theory is developed for describing the motion of these organisms under gravity, and the predictions of the theory are compared with measurements on suspensions of paramecia in long, vertical columns. The observations that low temperatures (Moore, 1903) and bright sunlight (Fox, 1925) induce positive geotaxis in Paramecium are interpreted in the light of the theory. Finally, the possible importance of geotaxis in other motile micro-organisms such as bacteria and spermatozoa is considered.
The kinetics of gravitaxis and gravikinesis in Paramecium caudatum were investigated by employing (1) step transitions from normal gravity (1 g) to weightlessness (microgravity) and (2) turns of the experimental chambers from the horizontal to the vertical position at 1 g. The transition to microgravity left existing cell orientations unchanged. Relaxation of negative gravitaxis under microgravity took longer than 10 s and may be described by the time constant of the decay of orientation coefficients. Gravitaxis was started at 1 g by turning the experimental chamber from a horizontal to a vertical position. Gravitaxis activated rapidly during the turning procedure and relaxed to an intermediate level after the turning had stopped. Gravity-induced regulation of swimming speed (gravikinesis) at 1 g had reached a steady state after 1 min; at this point, gravikinesis counteracted the effects of sedimentation (negative gravikinesis). A step transition to microgravity initially reversed the sign of the gravikinesis (positive gravikinesis). The relaxation of this kinetic response was not completed during 10 s of microgravity. The data suggest that gravikinesis is functionally unrelated to gravitaxis and is strongly affected by the rate of change in acceleration. We present a model explaining why gravikinesis reverses sign upon the onset of a step from 1 g to microgravity.
Wild-type and the morphological mutant kin 241 of Paramecium tetraurelia showed improved orientation away from the centre of gravity (negative gravitaxis) when accelerations were increased from 1 to 7 g. Gravitaxis was more pronounced in the mutant. A correlation between the efficiency of orientation and the applied g value suggests a physical basis for gravitaxis. Transiently enhanced rates of reversal of the swimming direction coincided with transiently enhanced gravitaxis because reversals occurred more often in downward swimmers than in upward swimmers. The results provide evidence of a physiological modulation of gravitaxis by means of the randomizing effect of depolarization-dependent swimming reversals. Gravity bimodally altered propulsion rates of wild-type P. tetraurelia so that sedimentation was partly antagonized in upward and downward swimmers (negative gravikinesis). In the mutant, only increases in propulsion were observed, although the orientation-dependent sensitivity of the gravikinetic response was the same as in the wild-type population. Observed swimming speed and sedimentation rates in the wild-type and mutant cells were linearly related to acceleration, allowing the determination of gravikinesis as a linear (and so far non-saturating) function of gravity.
The motion of swimming micro-organisms that have a preferred direction of travel, such as single-celled algae moving upwards (gravitaxis) or towards a light source (phototaxis), is modelled as the continuous limit of a correlated and biased random walk as the time step tends to zero. This model leads to a Fokker-Planck equation for the probability distribution function of the orientation of the cells, from which macroscopic parameters such as the mean cell swimming direction and the diffusion coefficient due to cell swimming can be calculated. The model is tested on experimental data for gravitaxis and phototaxis and used to derive values for the macroscopic parameters for future use in theories of bioconvection, for example.
The archaic eukaryote unicellular microorganism, Paramecium, is propelled by thousands of cilia, which are regulated by modulation of the membrane potential. Ciliates can successfully cope with gravity, which is the phylogenetically oldest stimulus for living things. One mechanism for overcoming sedimentation is negative gravitaxis, an orientational response antiparallel to the gravity vector. We have postulated the existence of a negative gravikinesis in Paramecium, i.e. a modulation of swimming speed as a function of cellular orientation in space. With negative gravikinesis, an upward oriented cell actively augments the rate of forward swimming and depresses active locomotion during downward orientation. A brief outline of the gravikinesis hypothesis is given on a quantitative basis and experimental data are presented which have confirmed the major assumptions.
Negative gravitaxis of Paramecium almost disappeared in solutions having specific gravity about the same as that of the organisms (1.04). The taxis turned to positive in solutions of specific gravity 1.08. Using a drop shaft at the Japan Microgravity Center, Hokkaido (JAMIC) we examined how swimming behaviour in these media was modified by changing gravitational conditions before, during and after free-fall. Tracks of swimming cells recorded on videotape indicate that the swimming cells continued upward and downward shift depending on the specific gravity of the external medium under 1-g conditions and these vertical displacements disappeared immediately after the moment of launch. The effectiveness of changing gravity to induce displacement of the cells seems to depend on the orientation of the cells to gravity. These results suggest a corelation between vertical displacement of the cell through the medium and a gravitactic mechanism in Paramecium.
'Bioconvection' is the name given to pattern-forming convective motions set up in suspensions of swimming micro-organisms. 'Gyrotaxis' describes the way the swimming is guided through a balance between the physical torques generated by viscous drag and by gravity operating on an asymmetric distribution of mass within the organism. When the organisms are heavier towards the rear, gyrotaxis turns them so that they swim towards regions of most rapid downflow. The presence of gyrotaxis means that bioconvective instability can develop from an initially uniform suspension, without an unstable density stratification. In this paper a continuum model for suspensions of gyrotactic micro-organisms is proposed and discussed; in particular, account is taken of the fact that the organisms of interest are non-spherical, so that their orientation is influenced by the strain rate in the ambient flow as well as the vorticity. This model is used to analyse the linear instability of a uniform suspension. It is shown that the suspension is unstable if the disturbance wavenumber is less than a critical value which, together with the wavenumber of the most rapidly growing disturbance, is calculated explicitly. The subsequent convection pattern is predicted to be three-dimensional (i.e. with variation in the vertical as well as the horizontal direction) if the cells are sufficiently elongated. Numerical results are given for suspensions of a particular algal species (Chlamydomonas nivalis); the predicted wavelength of the most rapidly growing disturbance is 5-6 times larger than the wavelength of steady-state patterns observed in experiments. The main reasons for the difference are probably that the analysis describes the onset of convection, not the final, nonlinear steady state, and that the experimental fluid layer has finite depth.
Gravitaxis, gravikinesis, and gravitropism are different graviresponses found in protists and plants. The phenomena have been intensively studied under variable stimulations ranging from microgravity to hypergravity. A huge amount of information is now available, e.g. about the time course of these events, their adaptation capacity, thresholds, and interaction between gravity and other environmental stimuli. There is growing evidence that a pure physical mechanism can be excluded for orientation of protists in the gravity field. Similarly, a physiological signal transduction chain has been postulated in plants. Current investigations focus on the question whether gravity is perceived by intracellular gravireceptors (e.g. the Muller organelle of the ciliate Loxodes, barium sulfate vacuoles in Chara rhizoids or starch statoliths in higher plants) or whether the whole cell acts as a sedimenting body exerting pressure on the lower membrane. Behavioral studies in density adjusted media, effects of inhibitors of mechano-sensitive ion channels or manipulations of the proposed gravireceptor structures revealed that both mechanisms have been developed in protists and plants. The threshold values for graviresponses indicate that even 10% of the normal gravitational field can be detected, which demands a focusing and amplifying system such as the cytoskeleton and second messengers.
Gravitactic behavior of microorganisms has been known for more than a hundred years. Euglena gracilis serves as a model system for gravity-triggered behavioral responses. Two basic mechanisms are discussed for gravitaxis: one is based on a physical mechanism where an asymmetric mass distribution pulls the cell passively in the correct orientation and, in contrast, the involvement of an active sensory system. A recently developed high-resolution motion-tracking system allows the analysis of single tracks during reorientation. The results are compared to a model developed by Fukui and Asai (1985) which describes gravitaxis of Paramecium caudatum on the basis of a physical mechanism. Taking into account the different size, different density, different mass distribution as well as the different velocity, results of the adapted model description of Paramecium were applied to measured data of Euglena. General shapes as well as the time scale of the predicted reorientational movement compared to measurements were different. The analysis clearly rules out the possibility that gravitaxis of Euglena gracilis is based on a pure physical phenomenon, and gives further support to the involvement of an active reorientational system. In addition, it could be shown that cell form changes during reorientation, even in an initial period where no angular change was observed.
Die Vertikalbewegungen von Paramecium caudatum
- J Dembowski
Dembowski, J. 1931 Die Vertikalbewegungen von Paramecium caudatum. Archiv fur Protistenkunde 74, 153–187.
Some facts concerning geotropic gatherings of Paramecium
- A Moore
Moore, A. 1903 Some facts concerning geotropic gatherings of Paramecium. Am. J. Physiol. 9, 238–244.
The biased random walk and the analysis of microorganism movement
- A M Roberts
Roberts, A. M. 1975 The biased random walk and the analysis of microorganism movement. In Swimming and Flying in Nature (ed. T. Y.-T. Wu, C. J. Brokaw & C. Brennen), vol. 1, pp. 377–394. Plenum Press.