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Table of Contents

Molecular Phenomena in Flipper Formation and Limits of Miniaturization

Table of Contents

Introduction

The goal of the following report is to explore the intricacies of marine animals’ flippers at a molecular level. Three main topics will be addressed throughout the analysis: the phenomena underlying the formation of flippers from embryos, the molecular processes involved in the fin-to-limb transition from fish to tetrapods, and the limits to the miniaturization of flippers. The development of flippers from an embryo can be explained by both the reaction diffusion patterns theory, first introduced by the mathematician Alan Turing, and by the positional information theory, introduced by Lewis Wolpert. Reaction-diffusion patterns describe not only the formation of flippers and fins, but also many other patterns encountered in nature, such as the spots on the skin of a giraffe. This theory is based on chemical reactions involving an inhibitor and an activator. The positional theory, however, explains that complex structures are created by polarities that were in the tissue initially. Further in the report, the theory of the fin-to-limb transition is explained by the fact that, in bony fishes, the formation of fins and limbs undergo extremely similar processes as they are both powered by positive feedback loops of proteins. In the last section, the limit of flippers’ miniaturization, the behavior of large aquatic animals is compared with that of very small organisms that live in water. The differences in their behavior are attributed to the differences in the fluid behavior, which is rationalized by Navier-Stokes equations and the Reynolds number. Flippers are then compared to the undulating membrane of unicellular organisms, which is considered to be their absolute limit of miniaturization.

Reaction-diffusion Pattern and its Part in the Formation of Limbs

Alan Turing, a British mathematician, provided an answer to the question of how structures and shapes of organisms come to be (Green and Sharpe, 2015). The solution he proposed is known as reaction-diffusion, also referred to as RD (Green and Sharpe, 2015). Turing hypothesized that certain chemicals could control different cells based on their concentration. These signaling molecules are called morphogens. He came up with a set of two mathematical equations that model the patterns produced as both the concentrations of the chemicals and the speed at which they diffuse varies (Ellison, 2019). The reaction-diffusion equations are represented generally in the following form:

∂_{t} q=Ḏ∇^2 q + R(q)

(1)

where q (x,t) represents the unknown vector function, is a diagonal matrix of diffusion coefficients, and R represents all local reactions (Wooley et al., 2017).

Although his paper details six types of spatial patterns, the most relevant to embryological development is the stable stripe/spot version with a regular periodicity (Green and Sharpe, 2015). His discovery and the idea that diffusion – increased disorder – could create its own pattern were met with skepticism. The morphogens initially proposed by Turing can be considered as local activators paired with lateral inhibitors (Green and Sharpe, 2015). These activators and inhibitors act both on each other and on themselves. Because activators have a positive effect on their own production, even small molecular fluctuations can result in the activator having higher concentrations at certain positions, which continue to rise due to the positive-feedback loop (Green and Sharpe, 2015).

The increased concentration of the activator increases the production of the inhibitor. The inhibitor diffuses at a higher speed than the activator, which allows the activator peak to stabilize itself since the faster diffusion of the inhibitor away from the peak results in lower repressive levels. As a result of this inhibitor diffusion, the regions neighboring the peak experience an increase in concentration of inhibitor, resulting in repressed activator levels. At distances even further from the peak, the inhibitor is no longer strong enough to repress the level of activators, leading to the formation of new peaks on either side (Green and Sharpe, 2015). As a result, the observed pattern is spatially periodic, which means there is an approximately equal amount of distance between each activator concentration peak (Fig. 1).

Fig. 1 Graph of concentration vs space of a Turing RD system. [Adapted from Green and Sharpe, 2015]

In summary, reaction-diffusion systems are complex networks of molecules controlled by positive and negative feedback cycles. The activation and inhibition of the molecules along with the dissimilar diffusion rates result in a chemical composition that is stable and non-uniform over a spatial domain (Newman and Bhat, 2007).

Turing’s reaction-diffusion is an important consideration in many areas of developmental biology, including the patterning and pigmentation of animal skin, hair follicles, teeth, and the formation of limbs (Newman and Bhat, 2007). There is strong evidence, obtained from a series of manipulative experiments along with computer modelling, that the patterning of digit specification in the vertebrate limb development is a Turing RD system (Green and Sharpe, 2015). The limb bud has a periodic gene expression pattern presaging the digits, which shows them as very regular in width and length. Despite what is indicated by the Turing pattern, it is evident that the size of digits does vary slightly. This suggests that digit formation is driven by more than one patterning mechanism (Green and Sharpe, 2015).

Fig. 2 Reaction diffusion acting in parallel with PI in the developing limb bud. [Adapted from Green and Sharpe, 2015]

The Turing system determines whether a cell becomes a digit cell or an interdigit cell while the subsequent decision of whether to become a big digit 3 or a small digit 1 is driven by a positional information-based system such as the Shh gradient (Fig. 2). The Shh gradient is an example of a morphogen concentration gradient. The morphogen Shh, or Sonic Hedgehog, plays a role in cell growth, cell specialization, and patterning of the body (“SHH gene- sonic hedgehog” | MedlinePlus, 2010).

The idea of positional information, also referred to as PI, is an alternative to RD and was proposed by Lewis Wolpert, an engineer (Green and Sharpe, 2015). Wolpert was aiming to describe a system that relies on using earlier polarities across the tissue to create more complex patterns downstream rather than a self-organising system (Green and Sharpe, 2015). Wolpert suggested that differences in morphogen concentrations could be gradual enough that varying positions could be defined based on their differences in concentration (Wolpert, 2016). The idea of PI introduces an interpretation step, rejecting the idea of the direct coupling of morphogen distribution and resulting pattern. This interpretation step supposedly allows a monotonic morphogen concentration gradient to produce any pattern, periodic or not (Fig. 3). Essentially, the idea of positional information implies that morphogen concentrations act as positional coordinates. The local concentration of morphogen is interpreted by the cells, leading them to whichever fate choice the concentration of morphogen corresponds to (Green and Sharpe, 2015).

Fig. 3 Graph of concentration vs space for Wolpert’s PI system. [Adapted from Green and Sharpe, 2015]

The preceding case of patterning of digit specification in the vertebrate limb is an example of RD and PI acting in parallel with each other, however that is not their only method of interaction. Reaction diffusion can act both upstream and downstream of PI. Once again, the formation of the limb is used as an example to illustrate RD acting downstream of PI (Green and Sharpe, 2015). Certain biological cases of Turing patterns demonstrate a higher level of control, such as stripes requiring a longer wavelength in specific regions of tissue than others, than the simple Turing patterns which are regular and periodic across the entire spatial domain. In the development of the limb, the periodic pattern of the digit gene expression stripes develops as a radial arrangement with digits fanning out to satisfy the shape of the paddle-like distal limb rather than as a series of parallel stripes (Green and Sharp, 2015). Since the wavelength at the proximal end of the digits is shorter than the wavelength at the distal end, this fanning out of digits contradicts reaction-diffusion as it goes against the constant wavelength rule. Because of this, the reaction diffusion patterning must be controlled by a positional information system where the positional signal varies along the proximodistal axis of the limb (Green and Sharpe, 2015). In other words, the wavelength of the Turing pattern in the limb bud tissue is determined by its position along the proximodistal axis, therefore RD acts downstream of PI (Fig. 4).

Fig. 4 RD acting downstream of PI, causing the periodically patterned digits of mouse limbs to splay out distally. [Adapted from Green and Sharpe, 2015]

Pectoral Fin Development of the Catshark as a Turing Mechanism

A study done by Onimaru et al. (2016) explores evidence and conducts experiments to show that the skeletal patterning of the pectoral fin of the Scyliorhinus canicula, otherwise known as the catshark, is controlled by a deeply conserved Turing mechanism implemented by BMP, SOX9 and WNT proteins, the same mechanism proposed to control mouse digit patterning. In the fins of catshark, the distal nodular elements emerge from a periodic spot pattern of Sox9 expression, which differs from the stripe patterning of mouse digits (Onimaru et al., 2016). Sox9 is a protein encoding gene which makes proteins that are critical during embryonic development (“SOX9 gene- SRY-box 9” | MedlinePlus, 2016). The researchers use a computer model to show that Sox9 gene expression in catshark fins can be explained by the Bmp-Sox9-Wnt network with altered spatial modulation. Experimental disturbance of Bmp and Wnt signaling in catshark embryos results in skeletal alterations which correspond to the effects predicted by the computer simulation (Onimaru et al., 2016). Ominaru et al. demonstrate that the spot-like Sox9 pattern can be explained by the BSW model with slight modifications. The BSW model refers to the Turing network formed by regulatory interactions between BMP, SOX9, and WNT, which creates a periodic molecular pre-pattern specifying the positions of the digits in the mouse limb (Onimaru et al., 2016).

The researchers discovered that the first periodic expression of Sox9 in the catshark pectoral fin is a distal row of spots along the anterior-posterior axis. To determine this, they used optical projection tomography, or OPT, to examine a time course of Sox9 expression (Fig. 5). In Fig. 5-iii, a curved row of spots is depicted. The spots are initially more continuous in the posterior region, but eventually break up into spots (Fig. 5iv and v).

Fig. 5 Spot pattern from OPT scans of Sox9 expression in catshark pectoral fin buds at stages 29-30. [Adapted from Onimaru et al., 2016] The corresponding lower panels indicate (in red) which part of the future skeleton is represented by the Sox9 pattern above. Scale bars, 100 μm.

Subsequently, the authors examined expression of Bmp and Wnt related genes in the catshark pectoral fin buds to determine if the patterning of the S. canicula pectoral fin was controlled by a Turing system homologous to that of the mouse digit. If the system controlling the catshark pectoral fin was similar to the system controlling mouse digit patterning, Bmp and Wnt might be active in a pattern out-of-phase with Sox9, as shown in Fig. 6 (Onimaru et al., 2016).             

Fig. 6 Generic Turing network in mouse digit patterning with out-of-phase expression patterns of Bmp and Wnt with Sox9. [Adapted from Onimaru et al., 2016]

Onimaru et al. observed that the protein Bmp2 was only active in the edges of the distal fin, contrary to mice, in which it expresses the strongest out-of-phase pattern with Sox9. They discovered that Bmp4 expressed a pattern complementary to Sox9 instead. They then determined that Wnt5b had an out-of-phase pattern with Sox9, similar to the patterning of the mouse limb bud. Overall, the researchers concluded that the relationship between Bmp, Sox9, and Wnt is conserved from fish to mammals (Onimaru et al., 2016).

Using a computational model, Onimaru et al. obtained a time course of pectoral fin morphologies, created a series of two-dimensional triangular meshes, and calculated hypothetical tissue trajectories to verify that a BSW Turing network could replicate the early spot pattern of the catshark fin (Fig. 7). They compared the results of the computational model with real tissue data they obtained from carbon-particle-based fate mapping. By comparing these results, the researchers developed a realistic computational growth map with the asymmetrical growth along the anterior-posterior axis of the S. canicula fin being consistent with the fate maps of chick limb buds (Onimaru et al., 2016).

Fig. 7 Construction of a fin growth model. [Adapted from Onimaru et al., 2016]

The researchers then explored whether the BSW model could also describe the catshark distal fin elements. They used partial differential equations to determine that the pattern formed spots instead of stripes when the production of Wnt significantly exceeded the production of Bmp. Subsequently, Onimaru et al. (2016) simulated the situation in the fin growth model and discovered that a uniform distribution of spots appeared which bore no resemblance to the actual Sox9 expression patterns.

The authors hypothesized that distal Hox genes and fibroblast growth factor signaling, also referred to as FGF signaling, could be factors in shaping the Sox9 expression into the curved row of spots at a certain distance from the distal fin edge. Through computer modelling and experimentation with FGF, Onimaru et al. (2016) discovered that the row of Sox9 expression spots appear to be positioned by FGF signaling.

Further, the researchers tested inhibitions of Bmp and Wnt (Fig. 8). To begin, they performed numerical simulations as well as experiments with embryos, which showed that S. canicula embryos treated with Bmp inhibitors depicted a loss of some or even all Sox9 expression spots. Long-term treatments of embryos with Bmp-inhibitor were carried out and it was observed that the long-term treatment sometimes resulted in the expansion of the apical ectodermal ridge-like structure and the width of the pectoral fin buds (Onimaru et al., 2016). This implies that the previously reported negative effect of Bmp on the chick AER is conserved in the fin buds of S. canicula.

Fig. 8 Results of Bmp and Wnt inhibition from computer model (in silico) and from embryonic experiments (in vivo). [Adapted from Onimaru et al., 2016]

The researchers proceeded to examine Wnt inhibition by decreasing the Wnt production term in the model. The Sox9 expression spots became partially fused into continuous regions, and the spots that did form were larger than in the control simulation. They confirmed this effect by treating the embryo with a Wnt inhibitor, which also resulted in a pattern of partial or complete fusion of the Sox9 expression spots into a continuous domain parallel to the distal fin edge of the catshark. The response to both inhibitions in the catshark fin bud is similar to the response of the same inhibitions in the mouse limb bud (Onimaru et al., 2016).

The results of all the above experiments and simulations taken together provides evidence that the distal elements of S. canicula pectoral fins and mouse digits share a deeply conserved Turing system. Both species demonstrate similar interactions between Bmp, Sox9, Wnt, and Fgf. The results of this study imply that the broad morphological diversity of distal fin and limb elements arose from the spatial re-organization of a conserved Turing mechanism.

Fin-to-limb-to-digits Transition

The increasing presence of tetrapods alive today was initially signified by a restructuring of the earth’s ecosystem in the Late Devonian extinction (Stewart et al., 2017). Across all types of vertebrates, from terrestrial, to aquatic, to aerial environments, these animals are found abundantly. This advancement is caused not only by the diversification of limbs but also, the presence of digits. Found on the dial end of vertebrate paired pectoral and pelvic appendages, these endoskeleton segments are segmented, non-branching, and parallel (Stewart et al., 2017).

Fig. 9 During fin-to-limb transition, various anatomical changes happened. These include the loss of dermal rays and the presence of digits. For each, orientation follows anterior on left. [Adapted from Stewart et al., 2017] (a) Pectoral fin skeleton of Tiktaalik rosae, an elpistotegid fish. Fin contains dermal skeleton and endochondral skeleton. (b) Forelimb skeleton of Acanthostega gunnari, stem tetrapod. Limb shows a polydactylous pattern, found in many early limbs. (c) Forelimb skeleton of human. Limb shows pentadactyl pattern and mesopodial elements.

Previously, the expression of the specialized genes identified autopods independently, and the origin of digits was assumed to be caused by a new gene regulatory state in the distal limb bud mesenchyme. Favorably, where the limb bud endures late-phase expression of hoxa13 and hoxd13, digits were found to develop. Additionally, a lack of the protein coding gene hoxa11, primarily expressed in the proximal domain, was also noticed here (Stewart et al., 2017). However, after studying various bony fishes, it was found that the early fin and limb developments have extremely similar initiating molecular mechanisms (Schneider et al., 2013). Considering the limb, as its bud grows, the apical ectodermal ridge (AER) is formed by ectodermal cells. These cells secrete fibroblast growth fibers, a specific protein that maintains the limb’s growth along the proximo-distal axis, on top of specifying the distal limb’s domain. On the other hand, the patterns for developing limbs across the antero-posterior axis are articulated by posterior mesenchymal cells (Schneider et al., 2013) posterior growth (Schneider et al., 2013).

Fig. 10 A model for the evolution of the autopod of tetrapods. [Adapted from Schneider et al., 2013] Light blue – proximal Hox domain, blue – distal Hox domain, dark blue – overlap of proximal and distal domains, orange – AF territory. (top) Transition of AER into AF limits cell proliferation and constrains growth of fin endochondral mesenchyme. (middle) Hypothetically, in non-digited sarcopterygian, prolonged exposure to AER signals promotes cell proliferation, expanding distal region. (bottom) In tetrapods, AER promotes growth and expansion of limb mesenchyme.
Fig. 11 A model of pectoral fin development. [Adapted from Yano et al., 2012] As the fin bud develops across the distal axis, cell shape in AF changes from spherical and slender, to thin and polygonal. (a) AER (red) is crucial for fin mesenchyme outgrowth (straight arrow). (b) After the AER transitions to AF, the distal AF acts on outgrowth of AF (curved arrows). (c) Distal AF is distinguished from proximal AF by gene expression and tissue organization.

The further development of the AER differs between fins and limbs. For instance, in fins, an apical fold (AF) is created and develops dermal fin rays as the AER elongates whereas, in tetrapods, the AER persists, never converting to an AF (Schneider et al., 2013). Despite this, there exists now and, in the past, sarcopterygians that have fin rays and endochondral skeletons. This is explained by the hypothesis that a delayed AER to AF transition happened during the evolution of their limbs. Due to the extended exposure to AER signals, the fin rays are damaged by the increased endochondral bones. This results in an expanded endochondral portion of the skeleton of the fin. Furthermore, a study with zebrafish where the AF was removed during fin development resulted in distal elongation of the endoskeleton. The increased cell proliferation caused proximal radials to fuse and distal radials to appear (Schneider et al., 2013).

The fin-to-limb transition is also related to the structural components of the fin fold, specifically its support by actinotrichia fibrils. These are formed by collagens associated with non-collagen components, including And1 and And2, elastoidin proteins (Schneider et al., 2013). During the study presented above, the AF was successfully impaired because the zebrafish lost its actinotrichia after the elastoidin proteins were destroyed using morpholino (Schneider et al., 2013). This is a process where morpholino oligomer’s nucleotide analogs block transcription or proper splicing of the mRNA by binding short sequences at the transcription start site of the desired proteins (Ablain and Zon, 2016). Additionally, abnormalities such as expanded distal skeleton and increased frequency of polydactyl (“Polydactyly” | Boston Children’s Hospital, n.d.), a deformity where the hand has one or more extra fingers, in early tetrapods, can be explained by the overgrowth of the homebox, hoxd13a, due to the loss of And1 and And2 (Schneider et al., 2013).

Regardless of these conclusions, the developmental steps that cause the loss of actinotrichia, and evidently the AF, at this point, remain weakly understood. Various inconsistencies in observation arise. For instance, sarcopterygians have substantial dermal and endochondral bones in their fins. This rejects the expected result predicted by the overexpression of hoxd13a (Schneider et al., 2013). Conversely, the increase in hoxd13a expression strongly correlates to the reduction of the AF in zebrafish. These together suggest that the origin of digits, rising from the development of a distal compartment of sarcopterygian appendages, resulting from the delayed transition and eventual loss of the AF and its actinotrichia proteins, requires more experimental data (Schneider et al., 2013).

The claim is that if mesenchymal cells remained in a terminal position within the bud, due to their inability to migrate into the fin fold, their position would provost differentiation into endoskeleton (Stewart et al., 2017). First, the stylopod, comparable to the humerus and femur, forms. Followed by the formation of the zeugopod, comparable to the ulna and radius, or tibia and fibula. Eventually, the autopod forms, comparable to the wrist and ankle, and their digits. Supported by the reaction-diffusion phenomena presented by A.M. Turing in 1952 (Stewart et al., 2017), various models for the patterning of the limb skeleton were developed.

Previously, these Turing-type models of limb development depended on the ability of the pre-chondrogenic limb to self-organize. Along with the autopod’s nodules of cartilage and regularly spaced rods, this was the basis of numerous claims (Stewart et al., 2017). Common among them all was the hypothesis concerning evolutionary transformation, wherein genes deeply conserved throughout vertebrate phylogeny produced the components of these proposed patterning mechanisms. However, recently, the lack of experimental data was overcome. Two tetrapods, a mammal and a bird, and one cartilaginous fish, a shark, were studied (Stewart et al., 2017). Expectantly, their reasoning on digit patterning, unique to each animal, proved generalizations to be impractical.

Beginning with the mouse, Sharpe et al. found that where and when the regulator of cartilage cell differentiation, transcription factor Sox9, is expressed, depends on an interaction between two morphogens and the autopod (Stewart et al., 2017). These signaling molecules include Bmp2 and a member of the Wnt family. Together, these three factors interact dynamically through a substrate-depleting Turing-type process, named the Bmp-Sox9-Wnt (BSW) network (Stewart et al., 2017). Furthermore, while observing the embryonic pectoral fins of catshark, this network proved fundamental in the formation of the distal components of the endoskeleton. Despite this array of cartilage nodules, the proximal parallel rods of cartilage are not assisted by the BSW network. In comparison to its presence in the mouse, it can be concluded that this network was not involved in the patterning of the proximal skeletal elements, rather the digits only. The morphologically different structures in different animals can be attributed to the modification of the BSW network values (Stewart et al., 2017).

Moreover, this network produces a distal array of repeated endoskeletal elements found in paired fins and appendages, across jawed vertebrates and tetrapods. Regarding stem tetrapods, it becomes challenging to identify a periodic pattern in their distal-most endoskeleton (Stewart et al., 2017). This suggests that in early sarcopterygians, their paired appendages did not sustain the BSW network. At this point, the network worked elsewhere in the body, for instance, the median fins instead. This relocation of a skeletal patterning system from one fin to another has been seen before, frequently in adipose fins. Therefore, to clarify the contribution of the BSW network to the origin of digits, it must be tested for the patterning of the endoskeleton of other fins and limbs (Stewart et al., 2017).

To continue, Bhat, Newman, and colleagues, studied the case on chicken. They found that its limb skeletal patterning mechanism came from a direct interaction of proteins and an indirect interaction of cell surface receptors (Stewart et al., 2017). From the family of carbohydrate-binding proteins, Gal1a acts as an activator whereas Gal8 acts as an inhibitor in this multi scale “reaction-diffusion-adhesion” Turing-type mechanism. Specifically, this is a morphodynamic mechanism because the network depends on the movement of cells to form patterns. Despite most of the evidence supporting this two-galectin network comes from vitro and vivo manipulations of autopodial mesenchyme, there are studies concerning localization and manipulation that confirms this mechanism acts more proximally, patterning the stylopod and zeugopod (Stewart et al., 2017).

A mathematical model has been developed to represent this two-galectin network. Using experimentally defined parameters and variables, it predicts the number of foci of cartilage that will form (Stewart et al., 2017). Likely, the limb pattern depends on the capacity of the galectins competing for a common cell surface receptor in the limb-bud mesenchyme. Additionally, the varying levels of concentration of Gal1a and Gal8 during development have an effect. As the limb bud extends, down-regulation of Gal8 is observed in the apical mesenchyme. This is caused when transcription factors associated with limb development find binding sites caused by a non-coding motif (Stewart et al., 2017). These transcription factors include Meis1, responsible for determination of proximal limb elemental identities, Tcfcpl1, factor expressed in murine limb musculogenesis, Runx1, necessary for differentiation of chondroprogenitor cells to chondrocytes, and Runx2, required for chondrocytic maturation (Stewart et al., 2017). Now, when the presence of Gal1a decreases, an increasing number of cartilage elements is found: precisely one stylopod element, two zeugopodial, and multiple autopodial.

Limit of Flippers’ Miniaturization

Flippers are an extraordinary adaptation of marine mammals’ limbs that enable them to move efficiently in water. One could wonder: is there such a locomotion method for very small organisms? In other words, is there a limit to a flipper’s miniaturization? To answer this question, the concepts of Navier-Stokes Equations (“Navier-Stokes Equations” | Numberphile | YouTube, 2019) and Reynolds number (“Reynolds Number” | Numberphile | YouTube, 2019) must be explained.

Navier-Stokes equations are a set of partial differential equations that were derived by Claude-Louis Navier and George Gabriel Stokes in the 19th century. They are obtained by applying classical Newtonian mechanics to fluids. These equations are still not understood very well, and approximations often must be made. However, they are a good enough starting point for many problems that relate to fluid flow.

The first equation, colloquially known as “The small equation”, has the formula: ▽u where u is a vector representing the velocity of the studied fluid in every direction. The letter ▽ is the gradient, which represents the differentiation of each component of the vector along its own coordinate. This equation is the direct application of the law of conservation of mass: no matter how a fluid moves or changes its shape, its mass always remains the same (“Navier-Stokes Equations” | Numberphile | YouTube, 2019).

The second equation, known as “The big equation” or “The momentum equation”, originates from Newton’s Second Law (F=MA). The momentum equation is:

ρ {du\over dt}=-p+μ^2 u+ρF 

(2)

where du/dt is the derivative of the fluid’s speed with respect to time – acceleration – and ρ is the density of the fluid.  -p+μ2u+ρF represents the sum of all internal and external forces associated to the liquid. Internal forces are viscous forces and forces that are driven by differences in pressure. External forces can be many things; the most common one being the weight of the fluid. The thrust generated by the flippers of an animal is also considered as an external force (Acheson and Acheson, 1990).

Reynolds number is a number that aims to describe the viscosity of the flow in a given fluid (refer to Burkit, 2020 in Appendix) as it is the ratio of the inertial forces divided by the viscous forces (“Life at a low Reynold’s number” | Swizec Teller, 2013). It is defined by the equation Re=ρVL/µ, which can be derived from the momentum Navier-Stokes equation (“Reynolds Number” | NASA, n.d.). In this equation, ρ corresponds to the density of the fluid, v, to its velocity,  µ to its viscosity and L to the length of the object in the fluid. The equation can also be written by replacing the viscosity (µ) and the density (ρ), by the kinematic velocity (v) whose unit is the Stoke and which is defined as the ratio of the viscosity to the density. When the Reynolds numbers are high (>>1), inertia prevails whereas when the Reynolds numbers are low (<< 1), viscosity prevails and motion becomes more difficult (“Physics of Life- Life at a Low Reynolds Number” | ESFTV | YouTube, 2011). Since water is more viscous than air, Reynolds numbers in water are smaller than in air. As organisms get smaller, their length diminishes which makes the Reynolds number decrease even more. As a result, very small organisms that live in water are in a world dominated by viscosity. As an example, a person swimming in water would have a Reynolds number of the order of 104 whereas a bacterium moving in water would have a Reynolds number of the order of 10-4 (Purcell, 1977). Animals that swim at high Reynolds numbers move forward by giving a backward motion that provides thrust (“Physics of Life- Life at a Low Reynolds Number” | ESFTV | YouTube, 2011). However, at small Reynolds numbers, the fluid is way too viscous, and organisms end up at their starting point if they use this kind of motion. Since the liquid is very viscous, the organisms must carry it with them as they travel, forcing them to use a lot of their energy. This phenomenon is called “added mass”. To give an idea of the situation, for us to perceive the world the same way as unicellular organisms do in water, we would have to move in a fluid a lot more viscous than air or water such as roofing tar (“Physics of Life- Life at a Low Reynolds Number” | ESFTV | YouTube, 2011).

The reason why “reciprocal motion” (Purcell, 1977) does not work at small Reynolds numbers comes from the momentum Navier-Stokes equation.

Fig. 12 Navier-Stokes equation from a Newtonian perspective. [Adapted from “A brief introduction to the Navier-Stokes equations and problem”| Ravon | YouTube, 2018]

When the Reynolds number is small, which also means that the regime is dominated by viscosity, the acceleration disappears from the equation and only the part of the equation that describes the forces remains. As the acceleration is no longer present, the Navier-Stokes equation no longer depends on time, which means that the locomotion of an organism does not depend on its speed (Purcell, 1977). This is better known as the scallop theorem. Scallops can only produce reciprocal motion as they can only either open or close their shell. Their normal way of moving is by slowly opening their shell and closing it very quickly.

Fig. 13 Scallops’ locomotion. [Adapted from Ackerman, 2019]

However, at low Reynolds numbers, the scallop cannot move forward since its locomotion no longer depends on the opening and closing speed of its shell. Therefore, every time the shell opens and gives momentum to the scallop, the closing of the shell gives the same propulsion force but in the opposite sense, which makes the locomotion of the scallop impossible. Accordingly, organisms that live in a world governed by small Reynolds numbers absolutely need to develop another locomotion technique. What scientists have observed is that, at very small Reynolds numbers, organisms propel themselves by periodic movements instead of reciprocal motion (Tlusty, n.d.). At low Reynolds numbers, there are two main locomotion structures: cilia and flagella. These structures, both made of microtubules, enable movement since they are powered by periodical movement. These two structures are powered by ATP molecules: dynein proteins attract and release pairs of microtubules, causing the structure to bend and create motion. Cilia are short extensions of the cell that are found in unicellular organisms like Paramecium (refer to Staffero, 2014 in Appendix). Organisms usually have many of them. As illustrated in the hereinbelow picture, cilia work by giving a stroke with the cilia completely unfolded (effective stroke), followed by a stroke in the other direction with the cilia folded (recovery stroke).

Fig. 14 Locomotion of cells with cilia. [Adapted from Tlusty, n.d.]

Some small organisms such as Escherichia Coli (refer to Garcia-Ojeda, 2015 in Appendix) or Euglena (refer to Staffero, 2014 in Appendix) have flagella instead of cilia. A flagellum looks like a long tail that is outside of the cell membrane; it is a longer structure than a cilium. Organisms usually have two flagella, but they can also have one (refer to Staffero, 2014 in Appendix). As exemplified in the hereafter Fig., it behaves just as a screw that turns around a fixed axis and provides thrust.

Fig. 15 Locomotion with a flagella. [Adapted from Tlusty, n.d.]

If the flagella folds back on the cytoplasm, it creates an undulating membrane (“Protozoa exhibit many morphologies” | Tulane University, n.d.). This structure has the purpose of increasing the motility of the organism (“Physics of Life- Life at a Low Reynolds Number” | ESFTV | YouTube, 2011). The main difference with flippers resides on its way of action as it is powered by cyclical motion instead of reciprocal motion. Since, at low Reynolds numbers, the acceleration disappears, organisms must constantly move their locomotion structures to advance. From the moment they stop creating thrust, they stop moving. As undulating membranes are very similar to flippers and accomplish the same function, the limit of flipper miniaturization could be considered as being the undulating membrane itself. However, as stated before, the way of operation of these two structures varies a lot as they operate at very distinct Reynolds numbers.

Fig. 16 Representation of an undulating membrane. [Adapted from Walsh, 2011]

Conclusion

This study explains the origin of flippers and describes their operation on a small scale. Their formation, during the development of the animal, is described as a result of reaction-diffusion patterns, a series of positive and negative feedback loops controlling the concentrations of various molecules which govern the growth of the limb. For instance, the Shh morphogen gradient plays an important role in the formation of flippers as it decides whether a cell will become part of a digit or not. For a long time, the formation of digits was considered to be the result of a completely new regulatory gene in animals that possessed them, which has since been proven to be wrong. Instead, differences in the intensity, chronology, and placement of the expression of the hoxa13 and hoxd13 genes caused digits to appear during the Late Devonian Extinction. The limit of flipper miniaturization is influenced by the size of the animals. Very small animals move at extremely low Reynolds numbers, which reduces the importance of acceleration – and therefore time – in their movements. As a result, smaller animals, particularly unicellular organisms, live in a world governed by viscosity, where reciprocal movements similar to the movement of flippers are ineffective. Subsequently, smaller organisms employ a locomotion organ which does not need to take part in reciprocal movements, such as the cilia or flagella.  

Appendix

“A brief introduction to the Navier-Stokes equations and problem.”, 20 June 2018, Ravon, YouTube, https://www.youtube.com/watch?v=bSN8dgGJMX8.

“Cilia and Flagella.”, 14 April 2014, Linda Staffero, YouTube, https://www.youtube.com/watch?v=srGZu4Wvdzg.

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“Reynolds number explained.”, 23 April 2020, Aliya Burkit, YouTube, https://www.youtube.com/watch?v=PNqwqWH3QP8.

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