The younger flagellum sets the beat for Chlamydomonas reinhardtii.

Eukaryotes swim with coordinated flagellar (ciliary) beating and steer by fine-tuning the coordination. The model organism for studying flagellate motility, Chlamydomonas reinhardtii, employs synchronous, breaststroke-like flagellar beating to swim, and it modulates the beating amplitudes differentially to steer. This strategy hinges on both inherent flagellar asymmetries (e.g. different response to chemical messengers) and such asymmetries being effectively coordinated in the synchronous beating. In C. reinhardtii, the synchrony of beating is known to be supported by a mechanical connection between flagella; however, how flagellar asymmetries persist in the synchrony remains elusive. For example, it has been speculated for decades that one flagellum leads the beating, as its dynamic properties (i.e. frequency, waveform, etc.) appear to be copied by the other one. In this study, we combine experiments, computations, and modeling efforts to elucidate the roles played by each flagellum in synchronous beating. With a non-invasive technique to selectively load each flagellum, we show that the coordinated beating essentially only responds to load exerted on the cis flagellum; and that such asymmetry in response derives from a unilateral coupling between the two flagella. Our results highlight a distinct role for each flagellum in coordination and have implication for biflagellates' tactic behaviors.

Many single-cell organisms use tiny hair-like structures called flagella to move around. To direct this movement, the flagella must work together and beat in a synchronous manner. In some organisms, coordination is achieved by each flagellum reacting to the flow generated by neighbouring flagella. In others, flagella are joined together by fiber connections between their bases, which allow movement to be coordinated through mechanical signals sent between flagella. One such organism is Chlamydomonas reinhardtii, a type of algae frequently used to study flagellar coordination. Its two flagella ­ named trans and cis because of their positions relative to the cell's eyespot ­ propel the cell through water using breaststroke-like movements. To steer, C. reinhardtii adjusts the strength of the strokes made by each flagellum. Despite this asymmetry, the flagella must continue to beat in synchrony to move efficiently. To understand how the cell manages these differences, Wei et al. exposed each flagellum to carefully generated oscillations in water so that each was exposed to different forces and their separate responses could be measured. A combination of experiments, modelling and computer simulations were then used to work out how the two flagella coordinate to steer the cell. Wei et al. found that only the cis flagellum coordinates the beating, with the trans flagellum simply copying the motion of the cis. A direct consequence of such one-way coupling is that only forces on the cis flagellum influence the coordinated beating dynamics of both flagella. These findings shed light on the unique roles of each flagellum in the coordinated movement in C. reinhardtii and have implications for how other organisms with mechanically-connected flagella navigate their environments.

Effect of fluid elasticity on the emergence of oscillations in an active elastic filament.

Many microorganisms propel themselves through complex media by deforming their flagella. The beat is thought to emerge from interactions between forces of the surrounding fluid, the passive elastic response from deformations of the flagellum and active forces from internal molecular motors. The beat varies in response to changes in the fluid rheology, including elasticity, but there are limited data on how systematic changes in elasticity alter the beat. This work analyses a related problem with fixed-strength driving force: the emergence of beating of an elastic planar filament driven by a follower force at the tip of a viscoelastic fluid. This analysis examines how the onset of oscillations depends on the strength of the force and viscoelastic parameters. Compared to a Newtonian fluid, it takes more force to induce the instability in viscoelastic fluids, and the frequency of the oscillation is higher. The linear analysis predicts that the frequency increases with the fluid relaxation time. Using numerical simulations, the model predictions are compared with experimental data on frequency changes in the bi-flagellated alga Chlamydomonas reinhardtii. The model shows the same trends in response to changes in both fluid viscosity and Deborah number and thus provides a possible mechanistic understanding of the experimental observations.

Short-term memory effects in the phototactic behavior of microalgae.

Phototaxis, the directed motion in response to a light stimulus, is crucial for motile microorganisms that rely on photosynthesis, such as the unicellular microalga Chlamydomonas reinhardtii. It is well known that microalgae adapt to ambient light stimuli. On time scales of several dozen minutes, when stimulated long enough, the response of the microalga evolves as if the light intensity were decreasing [A. Mayer, Chlamydomonas: Adaptation phenomena in phototaxis, Nature, 1968, 217(5131), 875-876]. Here, we show experimentally that microalgae also have a short-term memory, on the time scale of a couple of minutes, which is the opposite of adaptation. At these short time scales, when stimulated consecutively, the response of C. reinhardtii evolves as if the light intensity were increasing. Our experimental results are rationalized by the introduction of a simplified model of phototaxis. Memory comes from the interplay between an internal biochemical time scale and the time scale of the stimulus; as such, these memory effects are likely to be widespread in phototactic microorganisms.

Cell size: Single cells illuminate the rules of cell size control.

A new study uses Chlamydomonas reinhardtii to understand how cell size homeostasis emerges from stochastic individual cell behaviors within a population. The authors find that a simple power law model was a poor predictor of cell size regulation; rather, it is better explained by a modified threshold model.

Exploring cell aggregation as a defense strategy against perchlorate stress in Chlamydomonas reinhardtii through multi-omics analysis.

Perchlorate (ClO4-) is a type of novel, widely distributed, and persistent inorganic pollutant. However, the impacts of perchlorate on freshwater algae remain unclear. In this study, the response and defense mechanisms of microalgae (Chlamydomonas reinhardtii) under perchlorate stress were investigated by integrating physiological and biochemical monitoring, transcriptomics, and metabolomics. Weighted gene co-expression network analysis (WGCNA) of transcriptome data was used to analyze the relationship between genes and phenotype and screen the key pathways. C. reinhardtii exhibited aggregate behavior when exposed to 100- and 200-mM perchlorate but was restored to its unicellular lifestyle when transferred to fresh medium. WGCNA results found that the "carbohydrate metabolism" and "lipid metabolism" pathways were closely related to cell aggregation phenotype. The differential expression genes (DEGs) and differentially accumulated metabolites (DAMs) of these pathways were upregulated, indicating that the lipid and carbohydrate metabolisms were enhanced in aggregated cells. Additionally, most genes and metabolites related to phytohormone abscisic acid (ABA) biosynthesis and the mitogen-activated protein kinase (MAPK) signaling pathway were significantly upregulated, indicating their crucial roles in the signal transmission of aggregated cells. Meanwhile, in aggregated cells, extracellular polymeric substances (EPS) and lipid contents increased, photosynthesis activity decreased, and the antioxidant system was activated. These characteristics contributed to C. reinhardtii's improved resistance to perchlorate stress. Above results demonstrated that cell aggregation behavior was the principal defense strategy of C. reinhardtii against perchlorate. Overall, this study sheds new light on the impact mechanisms of perchlorate to aquatic microalgae and provides multi-omics insights into the research of multicellular-like aggregation as an adaptation strategy to abiotic stress. These results are beneficial for assessing the risk of perchlorate in aquatic environments.

A PSII photosynthetic control is activated in anoxic cultures of green algae following illumination.

Photosynthetic hydrogen production from microalgae is considered to have potential as a renewable energy source. Yet, the process has two main limitations holding it back from scaling up; (i) electron loss to competing processes, mainly carbon fixation and (ii) sensitivity to O2 which diminishes the expression and the activity of the hydrogenase enzyme catalyzing H2 production. Here we report a third, hitherto unknown challenge: We found that under anoxia, a slow-down switch is activated in photosystem II (PSII), diminishing the maximal photosynthetic productivity by three-fold. Using purified PSII and applying in vivo spectroscopic and mass spectrometric techniques on Chlamydomonas reinhardtii cultures, we show that this switch is activated under anoxia, within 10 s of illumination. Furthermore, we show that the recovery to the initial rate takes place following 15 min of dark anoxia, and propose a mechanism in which, modulation in electron transfer at the acceptor site of PSII diminishes its output. Such insights into the mechanism broaden our understanding of anoxic photosynthesis and its regulation in green algae and inspire new strategies to improve bio-energy yields.

The flagellar length control system: exploring the physical biology of organelle size.

How cells build and maintain dynamic structures of defined size is currently an important unsolved problem in quantitative cell biology. The flagella of the unicellular green algaChlamydomonasprovide a highly tractable model system to investigate this general question, but while the powerful genetics of this organism have revealed numerous genes required for proper flagellar length, in most cases we do not understand their mechanistic role in length control. Flagellar length can be viewed as the steady state solution of a dynamical system involving assembly and disassembly of axonemal microtubules, with assembly depending on an active transport process known as intraflagellar transport (IFT). The inherent length dependence of IFT gives rise to a family of simple models for length regulation that can account for many previously described phenomena such as the ability of flagella to maintain equal lengths. But these models requires that the cell has a way to measure flagellar length in order to adjust IFT rates accordingly. Several models for length sensing have been modeled theoretically and evaluated experimentally, allowing them to be ruled out. Current data support a model in which the diffusive return of the kinesin motor driving IFT provides a length dependence that ultimately is the basis for length regulation. By combining models of length sensing with a more detailed representation of cargo transport and availability, it is now becoming possible to formulate concrete hypotheses to explain length altering mutants.

Mutualism-enhancing mutations dominate early adaptation in a two-species microbial community.

Species interactions drive evolution while evolution shapes these interactions. The resulting eco-evolutionary dynamics and their repeatability depend on how adaptive mutations available to community members affect fitness and ecologically relevant traits. However, the diversity of adaptive mutations is not well characterized, and we do not know how this diversity is affected by the ecological milieu. Here we use barcode lineage tracking to address this question in a community of yeast Saccharomyces cerevisiae and alga Chlamydomonas reinhardtii that have a net commensal relationship that results from a balance between competitive and mutualistic interactions. We find that yeast has access to many adaptive mutations with diverse ecological consequences, in particular those that increase and reduce the yields of both species. The presence of the alga does not change which mutations are adaptive in yeast (that is, there is no fitness trade-off for yeast between growing alone or with alga), but rather shifts selection to favour yeast mutants that increase the yields of both species and make the mutualism stronger. Thus, in the presence of the alga, adaptative mutations contending for fixation in yeast are more likely to enhance the mutualism, even though cooperativity is not directly favoured by natural selection in our system. Our results demonstrate that ecological interactions not only alter the trajectory of evolution but also dictate its repeatability; in particular, weak mutualisms can repeatably evolve to become stronger.

Light-regulated adsorption and desorption of Chlamydomonas cells at surfaces.

Microbial colonization of surfaces represents the first step towards biofilm formation, which is a recurring phenomenon in nature with beneficial and detrimental implications in technological and medical settings. Consequently, there is interest in elucidating the fundamental aspects of the initial stages of biofilm formation of microorganisms on solid surfaces. While most of the research is oriented to understand bacterial surface colonization, the fundamental principles of surface colonization of motile, photosynthetic microbes remain largely unexplored so far. Recent single-cell studies showed that the flagellar adhesion of Chlamydomonas reinhardtii is switched on in blue light and switched off under red light [Kreis et al., Nat. Phys., 2018, 14, 45-49]. Here, we study this light-switchable surface association on the population level and measure the kinetics of adsorption and desorption of suspensions of motile C. reinhardtii cells on glass surfaces using bright-field optical microscopy. We observe that both processes exhibit a response lag relative to the time at which the blue- and red-light conditions are set and model this feature using time-delayed Langmuir-type kinetics. We find that cell adsorption occurs significantly faster than desorption, which we attribute to the protein-mediated molecular adhesion mechanism of the cells. Adsorption experiments using phototactically blind C. reinhardtii mutants demonstrate that phototaxis does not affect the cell adsorption kinetics. Hence, this framework can be used as an assay for characterizing the dynamics of the surface colonization of microbial species exhibiting light-regulated surface adhesion under precisely controlled environmental conditions.

Phenotyping single-cell motility in microfluidic confinement.

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.

Reminiscences of Robert Paul Levine (1927-2022).

I present my personal reminiscence of Paul Levine-a highly innovative scientist who did seminal work in photosynthesis. He was among the first to initiate and use a genetic approach toward photosynthesis. He greatly helped in establishing the green unicellular alga Chlamydomonas reinhardtii as a powerful model system not only for understanding the function of the photosynthetic apparatus but also for studying its biogenesis and regulation. During the period he spent at Harvard, he made several ground-breaking contributions such as identifying and establishing the order of some components of the photosynthetic electron transport chain as well as determining their genetic origin. He trained many students and post-doctoral fellows several of whom later became prominent in this field and in other areas of plant science.

Nonlinear Phototaxis and Instabilities in Suspensions of Light-Seeking Algae.

The mechanism by which living organisms seek optimal light conditions-phototaxis-is a fundamental process for motile photosynthetic microbes. It is involved in a broad array of natural processes and applications from bloom formation to the production of high-value chemicals in photobioreactors. Here, we show that a population of the model alga Chlamydomonas reinhardtii exhibits a highly sensitive nonlinear response to light and demonstrate that the self-organization of cells in a heterogeneous environment becomes unstable as the result of a coupling between bioconvective flows and phototaxis.

Microfluidic Platforms Designed for Morphological and Photosynthetic Investigations of Chlamydomonas reinhardtii on a Single-Cell Level.

Chlamydomonas reinhardtii is a model organism of increasing biotechnological importance, yet, the evaluation of its life cycle processes and photosynthesis on a single-cell level is largely unresolved. To facilitate the study of the relationship between morphology and photochemistry, we established microfluidics in combination with chlorophyll a fluorescence induction measurements. We developed two types of microfluidic platforms for single-cell investigations: (i) The traps of the "Tulip" device are suitable for capturing and immobilizing single cells, enabling the assessment of their photosynthesis for several hours without binding to a solid support surface. Using this "Tulip" platform, we performed high-quality non-photochemical quenching measurements and confirmed our earlier results on bulk cultures that non-photochemical quenching is higher in ascorbate-deficient mutants (Crvtc2-1) than in the wild-type. (ii) The traps of the "Pot" device were designed for capturing single cells and allowing the growth of the daughter cells within the traps. Using our most performant "Pot" device, we could demonstrate that the FV/FM parameter, an indicator of photosynthetic efficiency, varies considerably during the cell cycle. Our microfluidic devices, therefore, represent versatile platforms for the simultaneous morphological and photosynthetic investigations of C. reinhardtii on a single-cell level.

Hyperosmotic stress-induced microtubule disassembly in Chlamydomonas reinhardtii.

BACKGROUND: Land plants respond to drought and salinity by employing multitude of sophisticated mechanisms with physiological and developmental consequences. Abscisic acid-mediated signaling pathways have evolved as land plant ancestors explored their habitats toward terrestrial dry area, and now play major roles in hyperosmotic stress responses in flowering plants. Green algae living in fresh water habitat do not possess abscisic acid signaling pathways but need to cope with increasing salt concentrations or high osmolarity when challenged with adverse aquatic environment. Hyperosmotic stress responses in green algae are largely unexplored. RESULTS: In this study, we characterized hyperosmotic stress-induced cytoskeletal responses in Chlamydomonas reinhardtii, a fresh water green algae. The Chlamydomonas PROPYZAMIDE-HYPERSENSITEVE 1 (PHS1) tubulin kinase quickly and transiently phosphorylated a large proportion of cellular α-tubulin at Thr349 in G1 phase and during mitosis, which resulted in transient disassembly of microtubules, when challenged with > 0.2 M sorbitol or > 0.1 M NaCl. By using phs1 loss-of-function algal mutant cells, we demonstrated that transient microtubule destabilization by sorbitol did not affect cell growth in G1 phase but delayed mitotic cell cycle progression. Genome sequence analyses indicate that PHS1 genes evolved in ancestors of the Chlorophyta. Interestingly, PHS1 genes are present in all sequenced genomes of freshwater Chlorophyta green algae (including Chlamydomonas) but are absent in some marine algae of this phylum. CONCLUSION: PHS1-mediated tubulin phosphorylation was found to be partly responsible for the efficient stress-responsive mitotic delay in Chlamydomonas cells. Ancient hyperosmotic stress-triggered cytoskeletal remodeling responses thus emerged when the PHS1 tubulin kinase gene evolved in freshwater green algae.

A constitutive stress response is a result of low temperature growth in the Antarctic green alga Chlamydomonas sp. UWO241.

The Antarctic green alga Chlamydomonas sp. UWO241 is an obligate psychrophile that thrives in the cold (4-6°C) but is unable to survive at temperatures ≥18°C. Little is known how exposure to heat affects its physiology or whether it mounts a heat stress response in a manner comparable to mesophiles. Here, we dissect the responses of UWO241 to temperature stress by examining its growth, primary metabolome and transcriptome under steady-state low temperature and heat stress conditions. In comparison with Chlamydomonas reinhardtii, UWO241 constitutively accumulates metabolites and proteins commonly considered as stress markers, including soluble sugars, antioxidants, polyamines, and heat shock proteins to ensure efficient protein folding at low temperatures. We propose that this results from life at extreme conditions. A shift from 4°C to a non-permissive temperature of 24°C alters the UWO241 primary metabolome and transcriptome, but growth of UWO241 at higher permissive temperatures (10 and 15°C) does not provide enhanced heat protection. UWO241 also fails to induce the accumulation of HSPs when exposed to heat, suggesting that it has lost the ability to fine-tune its heat stress response. Our work adds to the growing body of research on temperature stress in psychrophiles, many of which are threatened by climate change.

The Chlamydomonas bZIP transcription factor BLZ8 confers oxidative stress tolerance by inducing the carbon-concentrating mechanism.

Photosynthetic organisms are exposed to various environmental sources of oxidative stress. Land plants have diverse mechanisms to withstand oxidative stress, but how microalgae do so remains unclear. Here, we characterized the Chlamydomonas reinhardtii basic leucine zipper (bZIP) transcription factor BLZ8, which is highly induced by oxidative stress. Oxidative stress tolerance increased with increasing BLZ8 expression levels. BLZ8 regulated the expression of genes likely involved in the carbon-concentrating mechanism (CCM): HIGH-LIGHT ACTIVATED 3 (HLA3), CARBONIC ANHYDRASE 7 (CAH7), and CARBONIC ANHYDRASE 8 (CAH8). BLZ8 expression increased the photosynthetic affinity for inorganic carbon under alkaline stress conditions, suggesting that BLZ8 induces the CCM. BLZ8 expression also increased the photosynthetic linear electron transfer rate, reducing the excitation pressure of the photosynthetic electron transport chain and in turn suppressing reactive oxygen species (ROS) production under oxidative stress conditions. A carbonic anhydrase inhibitor, ethoxzolamide, abolished the enhanced tolerance to alkaline stress conferred by BLZ8 overexpression. BLZ8 directly regulated the expression of the three target genes and required bZIP2 as a dimerization partner in activating CAH8 and HLA3. Our results suggest that a CCM-mediated increase in the CO2 supply for photosynthesis is critical to minimize oxidative damage in microalgae, since slow gas diffusion in aqueous environments limits CO2 availability for photosynthesis, which can trigger ROS formation.

Strong confinement of active microalgae leads to inversion of vortex flow and enhanced mixing.

Microorganisms swimming through viscous fluids imprint their propulsion mechanisms in the flow fields they generate. Extreme confinement of these swimmers between rigid boundaries often arises in natural and technological contexts, yet measurements of their mechanics in this regime are absent. Here, we show that strongly confining the microalga Chlamydomonas between two parallel plates not only inhibits its motility through contact friction with the walls but also leads, for purely mechanical reasons, to inversion of the surrounding vortex flows. Insights from the experiment lead to a simplified theoretical description of flow fields based on a quasi-2D Brinkman approximation to the Stokes equation rather than the usual method of images. We argue that this vortex flow inversion provides the advantage of enhanced fluid mixing despite higher friction. Overall, our results offer a comprehensive framework for analyzing the collective flows of strongly confined swimmers.

The Papain-like Cysteine Protease HpXBCP3 from Haematococcus pluvialis Involved in the Regulation of Growth, Salt Stress Tolerance and Chlorophyll Synthesis in Microalgae.

The papain-like cysteine proteases (PLCPs), the most important group of cysteine proteases, have been reported to participate in the regulation of growth, senescence, and abiotic stresses in plants. However, the functions of PLCPs and their roles in stress response in microalgae was rarely reported. The responses to different abiotic stresses in Haematococcus pluvialis were often observed, including growth regulation and astaxanthin accumulation. In this study, the cDNA of HpXBCP3 containing 1515 bp open reading frame (ORF) was firstly cloned from H. pluvialis by RT-PCR. The analysis of protein domains and molecular evolution showed that HpXBCP3 was closely related to AtXBCP3 from Arabidopsis. The expression pattern analysis revealed that it significantly responds to NaCl stress in H. pluvialis. Subsequently, transformants expressing HpXBCP3 in Chlamydomonas reinhardtii were obtained and subjected to transcriptomic analysis. Results showed that HpXBCP3 might affect the cell cycle regulation and DNA replication in transgenic Chlamydomonas, resulting in abnormal growth of transformants. Moreover, the expression of HpXBCP3 might increase the sensitivity to NaCl stress by regulating ubiquitin and the expression of WD40 proteins in microalgae. Furthermore, the expression of HpXBCP3 might improve chlorophyll content by up-regulating the expression of NADH-dependent glutamate synthases in C. reinhardtii. This study indicated for the first time that HpXBCP3 was involved in the regulation of cell growth, salt stress response, and chlorophyll synthesis in microalgae. Results in this study might enrich the understanding of PLCPs in microalgae and provide a novel perspective for studying the mechanism of environmental stress responses in H. pluvialis.

Bacterial marginolactones trigger formation of algal gloeocapsoids, protective aggregates on the verge of multicellularity.

Photosynthetic microorganisms including the green alga Chlamydomonas reinhardtii are essential to terrestrial habitats as they start the carbon cycle by conversion of CO2 to energy-rich organic carbohydrates. Terrestrial habitats are densely populated, and hence, microbial interactions mediated by natural products are inevitable. We previously discovered such an interaction between Streptomyces iranensis releasing the marginolactone azalomycin F in the presence of C. reinhardtii Whether the alga senses and reacts to azalomycin F remained unknown. Here, we report that sublethal concentrations of azalomycin F trigger the formation of a protective multicellular structure by C. reinhardtii, which we named gloeocapsoid. Gloeocapsoids contain several cells which share multiple cell membranes and cell walls and are surrounded by a spacious matrix consisting of acidic polysaccharides. After azalomycin F removal, gloeocapsoid aggregates readily disassemble, and single cells are released. The presence of marginolactone biosynthesis gene clusters in numerous streptomycetes, their ubiquity in soil, and our observation that other marginolactones such as desertomycin A and monazomycin also trigger the formation of gloeocapsoids suggests a cross-kingdom competition with ecological relevance. Furthermore, gloeocapsoids allow for the survival of C. reinhardtii at alkaline pH and otherwise lethal concentrations of azalomycin F. Their structure and polysaccharide matrix may be ancestral to the complex mucilage formed by multicellular members of the Chlamydomonadales such as Eudorina and Volvox Our finding suggests that multicellularity may have evolved to endure the presence of harmful competing bacteria. Additionally, it underlines the importance of natural products as microbial cues, which initiate interesting ecological scenarios of attack and counter defense.

Emergent probability fluxes in confined microbial navigation.

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 organization in microbial systems. While most of our current understanding is based on bulk systems or idealized geometries, it remains elusive how and at which length scale self-organization emerges in complex geometries. Here, using experiments and analytical and numerical calculations, we study the motion of motile cells under controlled microfluidic conditions and demonstrate that probability flux loops organize active motion, even at the level of a single cell exploring an isolated compartment of nontrivial geometry. 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. 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 geometries guiding their time-averaged motion.