Metachronal Motion across Scales: Current Challenges and Future Directions

Synopsis

Metachronal motion is used across a wide range of organisms for a diverse set of functions. However, despite its ubiquity, analysis of this behavior has been difficult to generalize across systems. Here we provide an overview of known commonalities and differences between systems that use metachrony to generate fluid flow. We also discuss strategies for standardizing terminology and defining future investigative directions that are analogous to other established subfields of biomechanics. Finally, we outline key challenges that are common to many metachronal systems, opportunities that have arisen due to the advent of new technology (both experimental and computational), and next steps for community development and collaboration across the nascent network of metachronal researchers.

Introduction: what is metachronal motion?

Metachrony refers to a motion that is not synchronous (meaning “occurring at the same time”); it is a specific subcategory of asynchrony (meaning the opposite). Metachrony implies a sequence of similar events which occur in sequential order. Biologically, metachronal motion occurs when a series of (usually morphologically similar) appendages sequentially perform a cyclic, repeated motion at a fixed phase lag from one another. This creates a “metachronal wave,” whose frequency and wavelength are governed by the spacing of the appendages and the phase lag between them. Each appendage, as it moves, drags the surrounding fluid along with it. Together, the sequentially coordinated appendages can generate fluid flows that are categorically different than those that would be created by a single appendage, or by the same group of appendages moving synchronously. Indeed, in some contexts, synchronous motion cannot create fluid flow at all, but metachronal motion can (Takagi 2015). Metachronally coordinated appendages can also produce a steadier, more efficient flow than the same appendages using synchronous coordination (Ford et al. 2019).

Metachronal motion is used across a staggering spectrum of species and scales, from microns to centimeters (e.g., Fig. 1). Because of this great functional and morphological diversity, there has been little cross-specific comparison of the physics of metachronal motion—and therefore no opportunity to discover potential commonalities. As a group of researchers working on several disparate systems, all of which use metachronal motion to drive fluid flow, we believe these as-yet-unknown commonalities may help us to (1) gain a better understanding of the fluid dynamics of metachronal motion, (2) drive the development of new bioinspired technology, and (3) ask and answer fundamental biological questions. Here we outline the current state of the field and give a brief picture of both the obstacles and opportunities inherent to developing a more holistic view of metachronal motion.

Fig. 1.

Sample images and size scales of biological and robotic metachronal swimmers, including (A) peacock mantis shrimp (Odontodactylus scyllarus), modified from Garayev and Murphy (2021). (B) Mysid shrimp (Americamysis bahia), modified from Ruszczyk et al. (2021). (C) Larval lobate ctenophore (Bolinopsis vitrea) (unpublished). (D) Tomopterid worm (Tomopteris krampi), modified from Daniels et al. (2021). (E) Copepod nauplius (Bestiolina similis), modified from Lenz et al. (2015). (F) Ghost shrimp (Palaemonetes sp.) (unpublished). (G) Krillbot (self-propelling robot), Ford and Santhanakrishnan (2020).

Multiscaled, multifunctional metachrony

One notable aspect of metachrony is that the characteristic size of the organism body $L$ may be many times greater than the appendage length $\ell$⁠. The difference in scale between metachronal swimmers and their appendages poses unique challenges to studying and understanding this mode of locomotion. First, this difference in scale presents experimental difficulties because of the extremely large dynamic range needed to simultaneously measure kinematics and fluid flow at both the appendage scale and the body scale. This problem is particularly vexing in ctenophores, for which the ratio $L/\ell$ often exceeds 30 (Matsumoto 1991); it is more manageable for organisms with smaller $L/\ell$ values such as euphausiids and shrimp (Catton et al. 2011; Garayev and Murphy 2021). The multiscaled nature of metachronal swimming is also computationally challenging because of the high levels of grid refinement required to resolve the flow around appendages while still reserving computational resources for the organism scale flow. Simultaneously resolving the broad range of scales involved in the metachronal motion will require innovative laboratory techniques and state-of-the-art technology (as we discuss later).

Metachrony is not only multiscaled but also multifunctional. In general, metachronal motion creates tunable flow at scales much larger than the individual units. This flow often takes the form of a (primarily) unidirectional, steady current just above a row or “carpet” of appendages, the uses of which are many and varied. Metachronal motion can generate propulsive forces for swimming (Gray 1939; Matsumoto 1991; Kohlhage and Yager 1994; Alben et al. 2010; Murphy et al. 2011; Colin et al. 2020), create feeding currents (Colin et al. 2010), enhance respiration (Sensenig et al. 2010), clear waste (Wanner et al. 1996), sort particles (Ding and Kanso 2015), and even attract symbionts (Nawroth et al. 2017). The versatility of metachronal motion indicates that it may hold potential for bioinspired technology development, and its ubiquity across taxa suggests the presence of somewhat universal biophysical pressures that drive evolution and development. Because metachronal motion has not been studied with a “big-picture” lens, with a targeted focus on similarities and differences between systems, these interesting questions remain open.

Much of the foundational work on metachronal coordination has its origins in very small scales: those of cilia and flagella. Arrays of cilia are found in organisms both micro and macroscopic; they are used for locomotion, feeding, reproduction, and countless other tasks (Machemer 1972; Wanner et al. 1996; Davis et al. 2006; Fauci and Dillon 2006; Satir and Christensen 2007; Riisgård and Larsen 2010). Unlike most metachronal swimmers we consider here, cilia are typically arranged in two-dimensional “carpets” rather than one-dimensional rows. This adds a degree of complexity to the coordination strategy. The metachronal wave can not only be antiplectic (where the individual power stroke direction is opposed to the direction of metachronal wave propagation) or symplectic (vice versa) but also dexio- or laeoplectic (the wave propagates obliquely to the power stroke). Symplectic metachrony is typically used to transport large particles (Knight-Jones 1954). However, systems which must efficiently create fluid flow (for locomotion, respiration, or other purposes) are primarily antiplectic, even when coordination passively emerges from hydrodynamic coupling (Michelin and Lauga 2010; Ghorbani and Najafi 2017; Chateau et al. 2019). This also holds true at larger scales: most metachronal-swimming organisms use primarily antiplectic metachrony (Colin et al. 2010; Sensenig et al. 2010; Tamm 2014).

Because of the breadth and depth of previous work regarding metachronal coordination in cilia, we will not attempt to provide a full summary here (although we will discuss some aspects of ciliary coordination within the more general context of metachrony writ large). For a more in-depth review, the reader is directed to reviews by Wan (2018) and Brennen and Winet (1977), as well as the landmark article by Elgeti and Gompper (2013).

Constraints and commonalities: morphology, kinematics, and dynamics

Organisms that use metachronal motion display broad morphological diversity in appendage shape, appendage size, number of appendages, appendage spacing, and flexibility of the body and appendages. However, these morphological parameters can limit the way that appendages can move, both individually and collectively. They also produce some surprising similarities across otherwise-dissimilar taxa.

Synchronicity and spacing

Appendage spacing and flexibility/articulation can particularly constrain the achievable values of the stroke amplitude and phase lag between adjacent appendages. Murphy et al. (2011) characterized the ratio of appendage spacing (⁠$B$⁠) to appendage length (⁠$\ell$⁠) for a range of species using metachronal swimming and observed that $B/\ell$ was in a narrow range of 0.2 < $B/\ell$  < 0.65. For a row of rigid appendages, decreasing $B/\ell$ (by either decreasing $B$ or increasing $\ell$⁠) would effectively lower the maximum achievable stroke amplitude, since the range of each stroke is limited to the available space between appendages. Body and appendage flexibility, however, may relieve this constraint by enabling a larger range of appendage motion, as noticeably seen in ciliary motion (Brennen 1974). Appendage spacing can also play an important role in time coordination between appendages. For example, animals with a smaller $B/\ell$ ratio (e.g., crustacean nauplii) typically perform a metachronal power stroke, but a nearly synchronous recovery stroke. By contrast, animals with relatively larger appendage spacing (higher $B/\ell$⁠) tend to perform metachronal power and recovery strokes (Murphy et al. 2011).

When compared with synchronous stroking, metachronal stroking augments swimming speed (Alben et al. 2010; Ford and Santhanakrishnan 2020) and increases fluid transport in cilia (Khaderi et al. 2012; Ding et al. 2014; Milana et al. 2020). Metachronal motion with a 20–25% phase lag results in the highest average flux (Zhang et al. 2014; Granzier-Nakajima et al. 2020) and fluid momentum (Ford et al. 2019) compared with other phase lags, which may explain why many organisms perform both the power and recovery stroke metachronally (including euphausiids (Alben et al. 2010; Murphy et al. 2011), mysids (Laverack et al. 1977; Hessler 1985; Schabes and Hamner 1992), and Anaspidacea (Grams and Richter 2021)). However, as mentioned above, some animals swim using a “hybrid” or “incomplete” metachronal stroke, where the power stroke is metachronal, but the recovery stroke is nearly synchronous. These stroke kinematics yield increasing speed during the power stroke, followed by a sharp decrease in speed during the recovery stroke. Animals that rely on these stroke patterns for cruising swimming, such as amphipods (Boudrias 2002) and isopods (Alexander 1988) show a sinusoidal time variation in their overall swimming speed which mimics the “hop-and-sink” swimming behavior of organisms with only one set of beating appendages, such as daphniids (Skipper et al. 2019) and juvenile Artemia sp. (Williams 1994). This initial acceleration may be crucial for the escape responses of animals that use these stroke patterns for intermittent/burst swimming (Ford et al. 2021) rather than steady swimming, as seen in copepods (van Duren and Videler 2003) and stomatopods (Campos et al. 2012).

While these observations—that appendage spacing constrains stroke amplitude, and is related to the synchronicity of stroke kinematics—are intriguing, the physical principles underpinning them are still unclear. It seems that animals with larger body sizes and/or a large total number of appendages are likely to have closer spacing between appendages, and maybe more likely to exhibit hybrid stroke kinematics (Murphy et al. 2011). However, it is unclear whether this tendency is driven by biology, fluid dynamics, or both. A systematic understanding of the relationship between flexibility, appendage spacing, and temporal coordination (as well as overall body size) could inform future studies of other metachronal movers, as well as provide a set of universal design principles that help us to interpret results.

The function of flexibility

Metachronal swimming often takes the form of collective drag-based paddling: overall thrust is maximized when the shape and kinematics of the propulsor maximize drag during the power stroke and minimize drag during the recovery. For single or paired appendages, this allows power-stroke drag on the paddle to sufficiently overcome both the resisting drag on the animal’s body as well as the recovery-stroke drag on the paddle (Vogel 2020). However, this simple view cannot fully capture the hydrodynamics of metachronal motion, in which often-flexible appendages execute complex spatiotemporal coordination patterns. Close spacing often makes it difficult to quantify the flow around and among appendages; only a few studies have managed it (Lim and DeMont 2009; Murphy et al. 2013; Ford and Santhanakrishnan 2021), primarily to estimate generated thrust using the trailing wake behind the propulsors. While this can accurately provide approximations of whole-body thrust, it will not provide insight into the underlying mechanisms that create thrust at the propulsor surface. There is, therefore, a great need for experiments and laboratory techniques that focus on visualizing the small-scale flows at the scales of individual metachronal appendages. Recent advances in millimeter-scale velocimetry techniques (Gemmell et al. 2014), non-invasive quantification of pressure (Dabiri et al. 2014), and force/torque estimation techniques (Lucas et al. 2017) have together enabled initial investigations into the forces generated by metachronal appendages at these small scales. For example, Colin et al. (2020) examined the hydrodynamics of a diverse array of evolutionarily distinct metachronal swimmers from three different phyla (Ctenophora, Annelida, Arthropoda) and demonstrated that these animals rely on generating negative pressures along the propulsive surfaces to generate forward thrust (i.e., “suction thrust”). Suction thrust has been documented in a variety of larger organisms with higher propulsor-scale Reynolds numbers and flexible bodies (Colin et al. 2012; Gemmell et al. 2015, 2016). However, this strategy may serve additional higher order purposes in metachronal motion: relying on negative pressure, as opposed to high pressure pushing forces, facilitates metachronal waves and enables these swimmers to exploit readily produced hydrodynamic structures (Colin et al. 2020). Here, a subtle bend (<30% inflexion) appears to generate a cascade of hydrodynamic effects which enhance the formation of negative pressure regions around propulsors and ultimately lead to enhanced thrust and hydrodynamic efficiency.

An interesting feature of swimming and flying animal propulsors—metachronal or not—is that regardless of size scale or Reynolds number, most propulsors bend similarly both in terms of bending location (with an inflexion point at approximately 65% of the propulsor length) and bend excursion (approximately 27°) (Lucas et al. 2014). Recent data on metachronal swimmers across three phyla are in line with these values despite their lower Reynolds numbers (Colin et al. 2020), suggesting that metachronal swimmers may follow some of the same design principles of other flexible creatures with potential added benefits as discussed above. However, unlike most other swimmers and fliers, many metachronal swimmers have highly deformable bodies in addition to their flexible appendages. Some animals (e.g., annelids of the families Nereididae and Tomopteridae [Gray 1939; Daniels et al. 2021]) display body undulations in addition to the metachronal wave of their appendages; these deformations may interact to varying degrees (Daniels et al. 2021). Some families of ctenophores (cestids, thalassocalycids [Swift et al. 2009]) also display large-amplitude body deformation in addition to metachronal motion. Siphonophores are unique among metachronal swimmers in that their “stroke cycle” consists of a sequential series of bell contractions by independent nectophores, which are highly flexible (Costello et al. 2015). However, the complex hydrodynamic interactions between multiple highly flexible propulsors and their substrates are neither well-defined nor well-understood. A better understanding of the physics of multiple interacting flexible surfaces and bodies could not only elucidate the study of metachronal swimming but could also potentially shed light on fundamental fluid dynamics and fluid-structure interaction problems. Such an understanding would be greatly beneficial for the development of devices (bioinspired or otherwise) at the millimeter-to-centimeter scale. Further investigation of the universality of propulsor bending dynamics, as applied to metachronal motion, may also lead to unexpected discoveries.

Gait transitions and the stability/maneuverability tradeoff

In general, aquatic animals with multiple appendages (e.g., copepods) are highly maneuverable (Yen 2013). While many metachronal swimmers are also capable of acrobatic maneuvers such as rapid accelerations and tight turns (Gemmell et al. 2019; Niimoto et al. 2020), most work has thus far focused on steady-state forward and/or backward swimming (Alben et al. 2010; Murphy et al. 2011; Daniels et al. 2021). However, as in terrestrial systems, it is important to identify and understand coordination patterns in steady-state locomotion before examining turning (Jindrich and Full 1999; Full et al. 2002). To understand the performance limits of metachronal motion in terms of maneuverability and agility, it may then be wise to first more fully consider steady-state swimming.

For swimming, steady-state locomotion would seem to imply a constant average speed such that (for an animal moving in a straight line) all hydrodynamic forces and moments acting on the various appendages and body sections are in balance with the forces and moments due to buoyancy. However, as in terrestrial systems, steady-state swimming is frequently pushed away from equilibrium: nonuniformities and disturbances in the aquatic environment, animal morphology, and appendage motor control produce forces that are not perfectly aligned with the desired swimming direction. It is thus important to examine the stability of forward swimming: will a random perturbation result in forces and moments that tend to increase the perturbation and destabilize the animal, or will these emerging forces tend to reduce the perturbation and return the animal to its original state? This description of stability immediately implies a tradeoff between stability (robustness to perturbation) and maneuverability (fast responses that can steer an animal away from its current direction) (Alexander 1982; Full et al. 2002; Weihs 2002; Jing and Kanso 2013; Huang et al. 2015).

To our knowledge, the stability of metachronal swimming is not yet well explored. Metachronal maneuverability and the ability to transition between distinct modes of coordination or gaits are even more understudied. In terrestrial systems, unidirectional, steady-state locomotion is also characterized by gait transitions between speed ranges, such as the transition from walking to running in humans or from trotting to galloping in horses (Alexander 1982). However, there are no analogous categories for metachronal motion that translate across systems. While some studies have defined distinct metachronal gaits (such as the hovering versus forward swimming examined by Murphy et al. [2011] and the cruising versus high-speed gaits observed by Kohlhage and Yager [1994]), these have not been generalized across taxa. In fact, to date, there is no clear functional dependence that clearly links metachronal coordination parameters (frequency, wavelength, kinematics of power and recovery stroke) to animal swimming gait (typically delineated by overall speed and discrete shifts in energetics). A notable exception comes from the study of ciliated micro-organisms (Hamel et al. 2011; Wan 2018, 2020). For example, the coordination modes observed in biflagellates, while perhaps less complex than other metachronal arrays considered here, provide valuable insights and a starting point to begin to examine multi-appendage coordination. Specifically, the biophysical mechanisms and role of the fluid medium in facilitating biflagellar coordination have been explored both experimentally (Brumley et al. 2014; Wan et al. 2014; Wan and Goldstein 2016) and mathematically (Guo et al. 2018, 2021; Man and Kanso 2020). This work suggests that it is possible to reach multiple coordination modes by varying the intrinsic activity of each flagellum (appendage) and the coupling strength between neighboring flagella (Guo et al. 2018, 2021; Man and Kanso 2020). Further insights into these findings and whether they can be generalized to large numbers of appendages and “higher” organisms with muscle-powered appendages will certainly require additional modeling in parallel with laboratory experiments. This will also require common kinematic/dynamic comparison points, as discussed in later sections.

Opportunities and limitations of modeling approaches

Experimental studies using live organisms can offer valuable insight into how the kinematics of metachronal motion affects hydrodynamics and mechanical performance. However, time-intensive experiments are needed across a wide range of species to identify the generalized structure–function relationships and scaling principles underlying metachronal systems. Behavior-driven variation and non-repeatability are often a concern when handling live animals in laboratory settings, and exhaustive statistical testing and/or large sample sizes are frequently required to address these artifacts. Field studies of live organisms often require expensive instrumentation and control for multi-camera recordings and lighting. However, physical models (static and dynamic/robotic) and computational models inspired by biological systems can serve as practical alternative approaches to examine the roles of structural and kinematics parameters on hydrodynamics and mechanical performance of metachronal motion. In addition, modeling studies can be used to examine “what if?” questions that are difficult or impossible to probe in live animals. Modeling approaches have been used in myriad biomechanics studies to yield new physical insight in widely disparate locomotion strategies, from insect flight to fish swimming. We identify some of the opportunities and limitations of experimental and computational models relevant to metachronal motion below.

Physical (robotic) modeling

Physical models can be broadly classified as “true-to-scale” or dynamically scaled based on whether length scales are matched between the biological and model counterparts. Since the exact representation of many metachronal species would require small-size (<1 cm) robots that are challenging to create, scaled-up models can be used as an alternate investigative approach.

A common dimensionless number used in fluid dynamic scaling is the Reynolds number (Re), defined as the ratio of inertial force imparted by a moving structure on the ambient fluid to the viscous force imparted by the fluid on the moving structure. Matching Re between a biological system and the corresponding model analog guarantees that dimensionless fluid dynamic forces (lift, drag, and thrust) in both systems are identical. Tethered scaled-up experimental models have been used to examine the role of phase lag in fluid pumping using oscillating plate arrays (Larson et al. 2014), as well as the effects of varying Re and phase lag on the generation of horizontal and downward momentum (Ford et al. 2019). Self-propelling models, where force generated by appendage motion allows forward motion in a single direction, can provide more realistic approximations than tethered models. These robots have been recently used to examine low Re swimming near a substrate (Hayashi and Takagi 2020), the role of phase lag (Ford and Santhanakrishnan 2020), appendage spacing ratio (⁠$B/\ell$⁠), and hybrid stroke kinematics seen in stomatopods (Ford et al. 2021).

As discussed previously, the co-existence of multiple length scales in metachronal systems presents challenges for the design of physical models. Simultaneously matching Re across all the length and velocity scales in a metachronal system may be difficult if not impossible. For example, we can match the Re based on body length and swimming speed of krill in a scaled-up physical model being towed in a tank, but it may be difficult to also simultaneously match the Re of flow past the pleopods and the Re of flow through the pleopod setae. A particular challenge in self-propelling models used to evaluate free-swimming performance is the resistance offered by the cabling of motors needed to drive the appendages of a scaled-up model (Ford and Santhanakrishnan 2020), which can limit normalized swimming speed (body lengths per second) and advance ratio (body speed to appendage tip speed ratio) of the travel. Precision manufacturing techniques may be required to incorporate small-scale structural features in physical models. Appendage flexibility is also difficult to reproduce in a physical model, particularly when deformation is passive and not prescribed. Moderate flexibility in drag-based paddling has been incorporated using polycarbonate plates (Kim and Gharib 2011), but more sophisticated manufacturing is required to incorporate differential bending in recovery with flexible plates. Other challenges in robotics include scalability for many-appendage systems (e.g., remipedes and tomopterids), and accurate representation of soft-bodied organisms. Recent advances in soft robotics (Milana et al. 2020) provide a potential path forward to incorporate appendage and body flexibility in experimental models.

Numerical modeling

With the rise of mathematical biology over the past half-century, much has been gained from the examination of biological systems using tools at the interface of physics, mathematics, and biology. Numerical modeling allows us to develop insight and intuition on the role metachrony plays in limb coordination during propulsion, without performing invasive procedures on specimens or relying on otherwise difficult-to-procure data. Modeling approaches vary in complexity from reduced-order analytical models (Williams 1994; Takagi 2015) to high-fidelity computational fluid dynamics (CFD) simulations (Zhang et al. 2014), each having its own strengths and weaknesses. Analytical and CFD models offer complementary toolsets and language to explore the parameter space that drives metachronal systems (Granzier-Nakajima et al. 2020; Herrera-Amaya and Byron, submitted for publication), and are useful both to supplement laboratory experiments and to explore questions that are not experimentally accessible (e.g., a numerical model can remove inherent physical constraints or test “unoccupied” regions of the parameter space to gain insight into fundamental questions about evolution and adaptation). Numerical modeling approaches can also drive cycles of innovation: for example, the increased interest in and utility of CFD for biological systems (which are often more flexible and deformable than engineered devices) has in turn driven the development of numerical algorithms to accurately and efficiently examine fluid–structure interactions within such systems (Cortez et al. 2005; Griffith and Patankar 2020). Additionally, the depth of scientific knowledge and the breadth of potential discovery are furthered when scientific disciplines can collaboratively study biomechanical systems in a holistic manner, where potential factors ranging from a neural organization (Zhang et al. 2014) and molecular architecture (Han and Peskin 2018) may drive the system. As tools in engineering and computation continue to advance, from new methods of data measurement (Katija et al. 2020) to the increasingly rapid development of machine learning (Brunton et al. 2020), numerical models can complement experimental tools to address important and timely challenges in biology.

Because of the sheer diversity of organisms that exhibit metachronal motion, there is an enormous parameter space that remains to be explored; numerical modeling represents a way to explore it quickly and efficiently. The range of scales involved, from the viscous-dominated regime of beating cilia to the inertial regimes of crustaceans and polychaetes, invites questions on the role of hydrodynamic efficiency at different Reynolds numbers (Guo et al. 2014). Simplified models have provided insight into the efficiency of crustaceans beating a small number of appendages at high (Alben et al. 2010) and low (Takagi 2015) Reynolds numbers (although these models did not account for hydrodynamic and fluid–structure interactions between adjacent appendages). Simplified models have also shed light on the efficiency of coordinating an extremely large number of cilia, usually by treating the ciliary sublayer as a continuous “envelope” (Blake 1971; Brennen 1974). However, in between these two extremes lies a complex regime: metachronal coordination of a moderate number of appendages. Many metachronal systems have on the order of tens of appendages, which have too many moving parts to treat independently and too few to generalize. Future investigations call for novel modeling approaches and high-fidelity simulations to explore the full spectrum of parameters, especially for these “in-between” cases.

As we have repeatedly discussed, the morphology of metachronal movers spans multiple length scales, and in some cases, the appendage size and body size differ by orders of magnitude. This presents multi-scale problems that may be computationally prohibitive to explore. It is still unclear how smaller-scale flows generated by individual appendages can be parametrized, or if this parametrization could be generalizable across metachronal systems. This is especially true for the moderate-appendage-number cases discussed above, for which CFD models are typically intimately tailored to a specific organism and cannot be ported to other metachronal movers without entirely rebuilding the simulation. Parametrization (i.e., closure modeling) of small scales could potentially promote generalizability in these models. However, when seeking to generalize or otherwise reduce the order of numerical models, parameter choice must be approached carefully. This is a persistent issue for all mathematical models and is particularly salient for biological systems that have not yet been fully studied experimentally.

Further challenges

We have previously discussed several major challenges inherent to the study of metachronal motion, including (1) its multiscaled nature, (2) its broad diversity in both form and function, (3) the tendency of metachronal swimmers to be flexible/deformable, and (4) the difficulty of using reduced-order analytical models for systems with a moderate number of appendages. However, there are currently several additional barriers to comparability and generalizability of results, access to needed data, and communication and collaboration between researchers. Below we outline three areas that must advance in order to facilitate the study of metachronal motion more broadly. First, we must establish uniformly defined parameters and reference variables that can be used across biological systems. Second, we must leverage recent technological advances in digital imaging and push forward with existing and new engineering projects which have broadly expanded the observability of organisms (e.g., deep-sea species). Finally, we must initiate (and support) community-building and data-sharing efforts to connect complementary datasets and skillsets between researchers across disciplines.

Standardization of nomenclature and reference parameters

Many laboratory and field studies have characterized stroke kinematics and mechanical performance of metachronal swimming in crustaceans, polychaetes, and ctenophores (Murphy et al. 2011; Colin et al. 2020; Daniels et al. 2021). However, the broad morphological and kinematic variation across metachronal movers, coupled with a lack of standardized reference parameters, makes it challenging to identify similarities (or differences) in the mechanical design of a metachronal system and concomitant effects on performance across taxa. The standardization of reference variables and parameters is therefore crucial for the development and validation of experimental and computational models aimed at elucidating the structure–function relationships underlying metachronal propulsion and transport. While there are many more parameters than those listed here, three important categories include stroke kinematics, appendage structure, and swimming performance. Our goal is to provide standardized definitions and reference parameters for use in the study of metachronal motion (Fig. 2).

Fig. 2.

Schematic representation of geometric parameters used to describe body and appendage position in metachronal motion. Posterior of animal is to the right; direction of travel is right-to-left. Lateral view (along body length) is shown in (A); magnified cross-section of a single biramous appendage (seen in many crustaceans) is shown in (B). The body is idealized to be a flat plate in (A). BA = body angle relative to global horizontal H; α = appendage angle; β = hinge angle for articulated appendages seen in many crustaceans; ℓ = appendage length; B = appendage spacing; PS = power stroke (counterclockwise direction in diagram, from anterior to posterior). For a flexible propulsor (FP, shown in blue), β can be defined as the angle between tangents drawn at the proximal and distal ends of the appendage. γ = abduction/adduction angle between the protopodite and endopodite of a biramous appendage (e.g., krill pleopod, sans setae).

Stroke kinematics

The appendage angle α is defined as the instantaneous angle between the long axis of the body and appendage, as viewed laterally. The time variation of this angle can be used to quantify stroke frequency (both for individual appendages and a mean across appendages) as well as phase lag between appendages. However, additional angles are needed for a full 3D reconstruction of stroke kinematics, including the bending angle β and the abduction/adduction angles γ. The bending (or hinge) angle is the angle between two tangents drawn at the most proximal and distal ends of the appendage. The abduction/adduction angle measures the angle that the appendages make with the dorsoventral plane. Finally, the body angle BA (that is, the angle of the long axis of the body with respect to the horizontal) is also required to differentiate between kinematics used for different gaits, such as hovering versus fast-forward swimming (Murphy et al. 2011). Other angles (e.g., the angle with respect to the vertical or with respect to the direction of locomotion) may also provide useful frameworks for the analysis of swimming kinematics. This metric may be less useful for animals without a well-defined “principal” body axis, such as ctenophores.

For cross-specific comparison, we must nondimensionalize angles and stroke durations. Although normalizing an angle with respect to its maximum value is the most straightforward approach, alternative scaling choices may also be needed when comparing across widely different morphologies. For example, the mean stroke angle can vary between appendages within the same animal (e.g., E. superba [Murphy et al. 2011]), which in turn can impact the flow generated by appendage motion and swimming performance. When quantifying phase lags between ipsilateral appendages, it is important to use a robust mathematical definition; using the peak-to-peak time separation in stroke angle profiles of appendages may be less than clear in circumstances where the power-recovery stroke is dramatically asymmetric. In animals with “hybrid” (metachronal-synchronous) stroke kinematics, it would be advisable to define two independent phase lags for both the power and recovery strokes.

Appendage structure

The size, shape, and number of stroking appendages vary widely across species using metachrony. Furthermore, nearly all biological appendages show some level of flexion during oscillatory motion. Appendage size can be broadly characterized via the dimensionless aspect ratio, defined as appendage length (or height) divided by width. For flexible appendages, such as ctenes or parapodia, we recommend defining aspect ratio with the appendage in an unbent state; for appendages that experience shape change, such as setal arrays or pinnules, we recommend reporting a range of values and reporting absolute (dimensional) quantities as needed. In general, highly flexible and deforming appendages are more difficult to characterize than hinged systems, requiring a larger number of parameters to fully describe. This is particularly true if the deformation is occurring in three dimensions. This is a significant challenge, particularly when comparing flexible systems to rigid or hinged systems. New nondimensional parameters and analysis frameworks may be needed to maximize insight from comparing such systems.

To describe appendage shape, an idealized characterization can be performed by comparing against the nearest geometry, such as trapezoidal pleopods in krill and rectangular pleopods in stomatopods. However, the idealization of appendage shape can be inadequate when attempting to isolate the role of structure, and theoretical approaches such as shape parameterization or morphospace analysis (Raup and Michelson 1965) may be required. The spacing ratio (⁠$B/\ell$⁠, discussed earlier) has been proposed as an important parameter in metachronal system design (Murphy et al. 2011) and is important to include in studies of metachronal motion. Although the bending angle also can be used to understand the role of appendage flexibility (Colin et al. 2020), dimensionless indices such as flexural stiffness (Kim and Gharib 2011) are recommended for comparison across species, especially in highly deformable systems that undergo continuous bending (e.g., ctenes and parapodia) rather than hinging at joints (e.g., pleopods of shrimp and krill). For certain systems, it may also be appropriate to further quantify the degree of spatial asymmetry in the stroke cycle, for example, by the ratio of flow-normal area in the power versus recovery strokes (Daniels et al. 2021; Herrera-Amaya and Byron, submitted for publication).

Swimming performance

Standard parameters such as normalized swimming speed (body lengths per second), body displacement per stroke (displacement efficiency), Froude efficiency (Dabiri et al. 2010) and advance ratio (Ellington 1984) may also be used for the study of metachronal motion. However, these parameters may need to be adjusted for the specific context of multi-appendage systems. In particular, the advance ratio (i.e., body speed relative to mean appendage tip speed) should be normalized by the number of stroking appendages (Murphy et al. 2011) to account for diversity in the latter variable across species. Hydrodynamic efficiency of oscillatory, lift-based locomotion (Walker 2002) is typically characterized using the dimensionless Strouhal number (Taylor et al. 2003). However, the Strouhal number is not readily adapted for drag-based locomotion (as in metachronal swimming) due to uncertainty in the definition of characteristic speed to use (Murphy et al. 2011). The wake speed has been useful in metachronal swimming studies, but questions remain as to the appropriate spatial location at which to extract the wake speed (Murphy et al. 2013; Ford and Santhanakrishnan 2020). The wave advance ratio, defined as the ratio of metachronal wave speed to body speed, may also be of use in comparing swimming performance across species. Characterizing the energetic cost of transport (Daniel 1985) may also elucidate why the metachronal propulsion strategy has been widely adopted across multiple taxa. This approach generally has not been included in studies of metachronal motion to date but could provide key insight into the efficiency and efficacy of metachronal motion.

Technological challenges and needed advances

Recent technological advances have significantly improved our ability to study metachrony across spatiotemporal scales. In particular, the rapid development of digital imaging technology has allowed scientists to capture behavior, kinematics, and fluid mechanics in flexible and scalable data formats and observation systems. High-speed, high-resolution imaging systems have expanded our capabilities to capture biological processes and behaviors that were once invisible to the naked eye. However, analysis of collected data remains nontrivial. The first step of this analysis is typically kinematics data extraction from video sequences, which often requires time-intensive feature tracking (Hedrick 2008; Schneider et al. 2012). Even when assisted by computer vision tracking algorithms (e.g., DLTdv [Hedrick 2008] or DeepLabCut [Mathis et al. 2018]), significant time is required by subject matter experts or trained analysts to convert digital video to meaningful data for comparative or mechanistic studies. Promising developments to address this bottleneck include leveraging multi-camera arrays to improve automated feature identification (Zong et al. 2018), resolving or reconstructing multi-dimensional features (Murphy et al. 2013), further improvement of machine learning-enabled image processing techniques (Hedrick 2008; Mathis et al. 2018), and availability of high-performance computing hardware to accelerate these methods.

Understanding metachronal swimming requires understanding the hydrodynamics involved. Quantification of the generated volumetric flow fields requires dedicated experiments and equipment, such as laser-assisted imaging for particle tracking or particle image velocimetry (PTV/PIV), other flow sensors, and multi-camera or plenoptic imaging arrays. Three-dimensional measurements of kinematics and hydrodynamics remain challenging to use and can be prohibitively expensive for researchers in both the laboratory and the field. The multiscaled nature of metachrony, as discussed above, also manifests explicit challenges for measurement: imaging across simultaneous spatiotemporal scales is required to capture the complex interactions between appendages while also quantifying the mechanics and movement of the entire organism. Imaging systems that can provide this multi-scale capability may require dynamic enhancements that can track an individual’s movements throughout the observations to enable adaptive imaging at smaller size scales. Systems like these have been demonstrated in microscopic planktonic organisms (Krishnamurthy et al. 2020); similar developments are needed for laboratory observations at larger scales.

Although it is important to address and enhance laboratory capabilities, we must also focus on how these systems function in their natural environment. Through this lens, we can enhance our understanding of the limits under which organisms have evolved to exploit metachronal locomotion. In other words, it is vital to incorporate ecology into our understanding of biomechanics: this is the relatively nascent field of ecomechanics. Expanding the “ecomechanics toolkit” should, therefore, be a high priority in studying metachrony. Investigation of in situ metachronal swimmers will enable us to ascertain metachrony’s full biological relevance by capturing metachronal gaits and transitions that may not be observable in the laboratory. While the latest digital imaging equipment and techniques may be implemented in controlled laboratory settings with relative ease, performing the same experiments in situ in aquatic environments poses significant and unique challenges. These challenges include the engineering challenge of modifying laboratory measurement systems and techniques for field use and limited access to study animals (driven by, e.g., limits on bottom time and working depths for SCUBA divers, the prohibitive expense of capable underwater robotic vehicles, and sparse or unpredictable distributions of target species).

To address these challenges, researchers have been deploying quantitative imaging systems both in field-based laboratories as well as in situ using SCUBA and robotic vehicle-based platforms (Dabiri et al. 2005; Katija and Dabiri 2008; Murphy et al. 2013; Gemmell et al. 2014; Katija et al. 2020). These developments have ranged from modifying and recombining off-the-shelf components to intensive multi-year engineering efforts which successfully integrated advanced imaging capability and multi-scale tracking on deep-diving remote autonomous vehicles (Katija et al. 2017; Yoerger et al. 2018; Katija et al. 2020). The Mesobot, a new class of underwater vehicle designed to autonomously track individual animals for more than 24 h, will enable simultaneous observations of organismal excursions (on the scale of hundreds of meters) and swimming behavior (centimeters) in an animal’s natural environment for the first time (Yoerger et al. 2018). These technologies have been successfully demonstrated on a wide range of organismal groups (e.g., ctenophores, siphonophores, medusae, krill, fish, larvaceans, and pteropods). Continuing advances in 3D imaging, non-invasive illumination, and improved observational platforms will enable observations of many others. The integration of artificial intelligence with vehicle control and navigation algorithms (Katija et al. 2021) could equip vehicles to autonomously search for and observe target organisms in the ocean, thereby expanding our observational capabilities even further.

Community-building across disciplines, systems, and skillsets

Understanding metachrony requires the use of multidisciplinary approaches from the computational, mechanical, chemical, and biological sciences. While our focus here is on metachronal swimming, other biomechanics and ecomechanics phenomena could be elucidated using the same methods. How can we connect a diverse range of researchers to study these problems, which (1) usually appear in non-traditional model organisms, (2) are often difficult to access and observe, and (3) typically require field observations as a baseline for behavior and capability? Researchers in the field tend to target specific organismal groups, while opportunistically sampling and observing non-target animals. How can we make this opportunistically collected data available to other researchers with different goals for comparative or synthesis studies? Open data repositories can be instrumental in connecting researchers in this context and can include a range of observation types. Shareable data may include digital images of animals in their environment, animal collections and microscopy data for tissue synthesis (e.g., to understand active/passive appendage actuation), quantitative measurements of animal behaviors in the laboratory, and many other data types.

Data repositories like this exist in many fields of biology (e.g., Morphosource [https://www.morphosource.org/] for 3D models of rigid organismal structures or xromm [https://xromm.rcc.uchicago.edu/] for X-ray footage of animal movement in the laboratory). However, an organismal-focused (rather than project-focused) repository that integrates a number of data types (e.g., imagery and video of organismal kinematics, tissue composition, and fluid flow in both laboratory and field environments) would be valuable not only for the study of metachronal motion, but for biomechanics in general and particularly for biomechanics/ecomechanics of marine organisms. Morphosource is built on a taxonomic tree, which enables easy navigation and determination of 3D model data quality and scope across different taxa. The Zoological Motion Analysis Portal (ZMAP, http://zmaportal.org) is an existing data repository for animal motion footage that allows researchers to upload and share various projects that include video data, calibrations, and other relevant metadata. Still, the taxonomy-based organization may be better able to facilitate opportunistic collaborations and data-sharing. With this organizational structure, different data types could be aggregated for individual species, enhancing value for metachrony researchers and the biomechanics community in general.

A centralized data repository for organismal biomechanics data could open existing distributed data sets to the broader research community. For example, the Monterey Bay Aquarium Research Institute’s Video Annotation and Reference System (VARS) contains 30+ years of video and synchronized environmental data from deep-diving remotely operated vehicles (Schlining and Stout 2006). Taxonomic experts have annotated thousands of hours of video footage, providing a searchable database for specific annotations of animals observed by the robotic vehicles. With more than 7 M annotations, 500 k framegrabs, and 4000 k concepts that include multiple observations of marine organisms swimming and feeding primarily in the Monterey Bay area (from the sea surface to 4,000 m deep), these data can be mined to generate clips of kinematics and behavior of rare deep-sea animals and added to the data repository. Other wildlife data repositories (e.g., Google’s Wildlife Insights, Cal Academy’s iNaturalist) and video data on YouTube of interesting animal behaviors and movements could also be leveraged for such a repository; in fact, data from some of these sources have already been used to derive novel insights in biomechanics (Lucas et al. 2017; Yang et al. 2017).

While video and image data can be costly data types to manage and host, new approaches in data storage, database design, and standards for metadata can mitigate these issues. The cost of data storage and hosting in the cloud has continually gone down while data download and upload performance has continually increased. If centralized data storage is prohibitive, decentralized approaches—where image and video data are hosted at various institutions and aggregated from a centralized database using APIs—have proven successful in the machine learning and computer vision communities (e.g., ImageNet [Deng et al. 2009] and FathomNet [Boulais et al. 2020]). Data standards, like Darwin Core Archive (Wieczorek et al. 2012), are accepted by data networks like OBIS (Grassle 2000) and IOOS (Snowden et al. 2019), and could also be applied here. Although constructing such a network to enable rapid ingestion and deployment of data would present a significant logistical/organizational challenge, an open data repository of this nature would be beneficial among the organismal biomechanics community, and could serve as a topic for a future research coordination network.

Summary and future directions

Here we have attempted to create a cohesive framework for the study of metachronal motion across an extremely diverse array of organisms. We have defined metachronal motion in the context of biology and included suggestions for the standard definitions of key parameters, investigative directions which could be ported from other subfields of biomechanics, and recommended approaches for community-building and data-sharing. We have also outlined some key challenges which are specific to the study of metachronal motion, and are distinct from the larger and more general discipline of biomechanics. These include the multiscaled nature of metachrony and the accompanying requirement to resolve spatiotemporal measurements across several orders of magnitude; the need to understand kinematic and dynamic constraints imposed by the close spacing of appendages that are often highly flexible; the current uncertainty surrounding how stability and gait transitions should be defined; difficulties in simultaneously matching all the relevant dimensionless parameters in robotic model studies; the non-monotonic variation in the difficulty of numerical modeling approaches (with a moderate number of appendages presenting a less tractable problem than either many or few appendages); and the limits imposed by available technology for data collection and analysis. However, we feel that there are also great opportunities presented by the recent impetus to unify researchers studying metachronal motion (Byron et al. submitted for publication). The adoption of data-sharing strategies such as those recommended here may facilitate rapid advancement of our understanding of this mode of locomotion, shedding light on why it is used across such seemingly dissimilar organisms and potentially providing pathways for the development of bioinspired design principles based on metachrony. The increasing interdisciplinarity of the scientific enterprise will encourage and hasten the development of collaborations between biologists, engineers, and mathematicians who are interested in and equipped to address the challenges outlined above. We look forward to the exciting new discoveries—and new questions—that will emerge as a result of these collaborations.

From the symposium “Metachronal coordination of multiple appendages for swimming and pumping” presented at the virtual annual meeting of the Society for Integrative and Comparative Biology, January 3–7, 2021.

Acknowledgments

The authors appreciate valuable discussion contributions from symposium speakers Ken Kiger and Eleanor Lamont and participants in the symposium complementary session. The symposium was supported by generous contributions from the Society’s Divisions of Comparative Biomechanics and Invertebrate Zoology, The Crustacean Society, and the Company of Biologists (#EA385).