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The locomotory apparatus and paraxial swimming in fossil and living marine reptiles: comparing Nothosauroidea, Plesiosauria, and Chelonioidea

The terrestrial origins of the diapsid Sauropterygia and Testudines are uncertain, with the latter being highly controversially discussed to this day. For only 15 Ma, Nothosauroidea lived in shallow-marine seas of the Triassic. Contrastingly, the pelagic Plesiosauria evolved in the Late Triassic, dispersed globally, and inhabited the oceans of the Jurassic and Cretaceous for approximately 135 Ma. Since the Cretaceous (~ 100 Ma), Chelonioidea, the modern sea turtles, have populated the oceans. All three groups evolved aquatic paraxial locomotion. Nothosaurs swam with their foreflippers, supported by the swimming tail. Plesiosaurs are the only tetrapods to have ever evolved four hydrofoil-like flippers. The plesiosaur flipper beat cycle has been debated for nearly two centuries. The different proposed locomotory styles (rowing, rowing-flight, underwater flight) are discussed in this review. A fourth gait that is employed by Carettochelys insculpta, which combines rowing and flying, is introduced. The osteology of the locomotory apparatus of nothosaurs and plesiosaurs is reviewed and compared to that of extant underwater-flying Chelonioidea. In conclusion, underwater flight remains the favoured locomotory style for plesiosaurs. Also, the review reveals that nothosaur locomotion has largely remained unstudied. Further, our understanding of joint morphologies and mobilities of the foreflipper in nothosaurs, plesiosaurs, and even recent sea turtles, and of the hindflipper in plesiosaurs, is very limited. It is crucial to the discussion of locomotion, to find out, if certain limb cycles were even possible, as evidence seems to point to the improbability of a rowing motion because of limited humerus and femur long axis rotation in plesiosaurs.


3
Comparable to Cetacea (Houssaye et al. 2015), Sauropterygian evolution covers a spectrum of gradual adaptations from shallow-marine benthic (Rieppel 1995) to pelagic lifestyles (Rieppel 2000;Wintrich et al. 2017a). These are accompanied by fundamental restructuring of the locomotory apparatus and thus the mode of locomotion. Three sauropterygian taxa, namely Placodontia, Nothosauroidea, and Plesiosauroidea, employed paraxial locomotion. The following text will focus on evaluating what we know about the locomotion of two of them, namely Nothosauroidea and Plesiosauria, by comparing them to a recent functional analogue within the reptiles, the Chelonioidea. Chelonioidea were chosen as functional analogue because they fly underwater. Sea turtles were given the favor over penguins because they do not have a reduced digital number and because they do not have the highly derived avian flight mechanism (i.e., the acrocoracohumeral ligament and the osteological complex that resolves around it) that penguins and other birds have (Baier et al. 2007).
Plesiosaurs lived in open marine habitats and to a minor extent in brackish, and fluvial habitats (Kear et al. 2006). They were all faunivorous, but showed quite different dietary preferences, as is suggested by their dentition and by fossil gastric contents (Sato and Tanabe 1998;Cicimurri and Everhart 2001;McHenry et al. 2005;O'Keefe et al. 2017). Foffa et al. (2014b) suggested that some pliosauromorphs might also be generalists. Filter feeding has also been recently proposed for an elasmosaur (O'Keefe et al. 2017). It is noteworthy that plesiosaur skeletons are often associated with gastroliths. Whether gastroliths had an influence on the plesiosaur's buoyancy, or whether they aided in food processing has not been clarified yet (see Wings 2007 for review; Schmeisser and Gillette 2009;O'Gorman et al. 2013O'Gorman et al. , 2014. Jurassic and Cretaceous plesiosaurs have been found on all of today's continents and were globally distributed in the Mesozoic (for review, see Bardet et al. 2014: 875, fig. 4).
Chelonioidea, like Plesiosauria, are a long-lived lineage of secondarily aquatic reptiles (Bardet 1994;Motani 2009;Vincent et al. 2011Vincent et al. , 2013. Similarly to plesiosaurs (see also Bardet 1994;Rieppel 1999;O'Keefe 2001a;O'Keefe and Carrano 2005;Ketchum and Benson 2010;Bardet et al. 2014), Chelonioidea are globally distributed although they are restricted to warmer waters as they are ectotherms. Dermochelys coriacea poses the only exception because it has an increased body core temperature (gigantothermic), lives in colder waters, and is the only recent truly pelagic sea turtle (Davenport et al. 2015).
Chelonioidea have inhabited the oceans since the middle Cretaceous (~ 100 Ma) or possibly since the Early Cretaceous (~ 145 Ma) (Evers and Benson 2019) to the present day. Presumably the behaviour of recent chelonioids resembles that of their fossil ancestors closely (Motani 2009). Juveniles of e.g., Caretta caretta hatch on a beach, head in the "frenzy" towards the ocean, and spend their first years in the open ocean. These years are called "lost years" by Carr (1952) because it was unknown where the hatchlings and juveniles went and how they lived. Juvenile chelonioids are carried by oceanic currents with sargassum rafts, which serve as shelters and supply them with food. The currents bring them to new habitats. As subadults, they migrate to  Rieppel (2000); b Shang and Li (2015) differ from a in that Pachypleurosauria and Nothosauria form a clade, after Shang and Li (2015); c Placodontia remain the most basal and Pistosauroidea the most derived sauropterygians, while Corosaurus-Cymatosaurus (usually found as basal pistosauroids) are at the base of a clade comprising Pachypleurosauria and Nothosauria as well, after Neenan et al. (2013) coastal waters (Frick et al. 2009). When they reach sexual maturity, they migrate hundreds of kilometres across the ocean to return to their birthplace, to mate and nest (Meylan 1982;Limpus et al. 1992).
Joints-The glenoid articulation of nothosaurs has not been closely studied so far in terms of degrees of freedom. Storrs (1993) suggests that the rather oval (long axis oriented in dorsoventral direction) glenoid fossa seems to exclude long axis rotation of the humerus which has an oval articulation surface with a pre-axially to post-axially directed long axis (Fig. 4f). Further, he suggested that nothosaurs show secondarily stiffened foreflippers (Storrs 1993), e.g., a stiffened  Iwabe et al. (2005) a as sistergroup to Lepidosauria and Archosauria; b as sistergroup to Archosauria; c as sistergroup to Lepidosauria, d as sistergroup to Crocodylia; e most recent phylogeny depicting Testudines interrelationships, after Crawford et al. (2015) elbow joint (Kuhn-Schnyder 1987). In general, nothosaur foreflipper mobility has not been investigated in detail.

Mode of locomotion
In nothosaurs, the long swimming tail probably served as main propulsive organ with which they swam by lateral undulation. Nonetheless, authors agree with Carroll and Gaskill (1985) that the foreflippers played an important role in propulsion since they are strongly morphologically (Carroll and Gaskill 1985;Kuhn-Schnyder 1987;Sues 1987;Storrs 1993) and histologically derived (Krahl et al. 2013;Klein et al. 2016). As opposed to foreflipper morphology, nothosaur hindflipper morphology is plesiomorphic, and it thus is assumed that hindflippers were employed in manoeuvering (Storrs 1993). This hypothesis was corroborated by Krahl et al. (2013) and Klein et al. (2016) by studying Nothosaurus long bones histologically. They showed that humeri, unlike femora, of large nothosaur species show a remarkable microanatomical specialization, i.e., the cortical bone is extremely reduced in thickness. The cortical thickness is comparable to that of aerial birds. This was interpreted as adjustments of the foreflippers to torsional forces, suggesting a complex movement cycle, possibly underwater flight (Krahl et al. 2013;Klein et al. 2016). Further evidence for this hypothesis is provided by nothosaur trace fossils described by Zhang et al. (2014).

Plesiosauria
Osteology-The locomotory apparatus of Plesiosauria is morphologically highly derived over that of the more basal sauropterygians, e.g., nothosaurs. Plesiosaurs have a greatly shortened tail (Taylor 1989;Wintrich et al. 2017a), with possibly a tail fin (for review see Smith 2013) and a greatly shortened trunk. The latter is secondarily stiffened by densely packed gastralia and much enlarged plate-like girdle elements (scapula, coracoid, pubis, ischium), similar Here the musculus coracobrachialis magnus inserts fibrocartilagenously (see Krahl et al. 2019 for more information) Not to scale. ef entepicondylar foramen, mp medial process, tr trochanter, tu tuberosity to the convergently evolved turtle plastron (Sues 1987;Taylor 1989;Storrs 1993). The dorsal portion of the scapula as well as the ilium are much reduced in size relative to the ventrally lying bones of the non-homologous pectoral and pelvic girdle, which are much enlarged (Frey and Riess 1982;Tarsitano and Riess 1982;Godfrey 1984).
Dorsally, plesiosaur humeri and femora have a processlike structure which is called tuberosity in the former and trochanter in the latter (Andrews 1910). The capitulum of both bones is generally round in cross section. Distally, both bones are increasingly dorsoventrally flattened and pre-axially to post-axially expanded (Andrews 1910) (Fig. 4c, d, g, h). How humeri and femora expand distally may vary across taxa: Sometimes the pre-axial margin is rather straight and the post-axial margin is expanded and curved post-axially (Druckenmiller and Russell 2008;Schumacher and Martin 2015;Delsett et al. 2016;Sachs et al. 2016) (Fig. 6a, d), sometimes the both the pre-and post-axial margin are expanded and curved pre-axially and post-axially (Hawkins 1840;Andrews 1910;Großmann 2006;Frey et al. 2017). The degree of expansion may also vary (Hawkins 1840;Andrews 1910;Großmann 2006;Druckenmiller and Russell 2008;Schumacher and Martin 2015;Delsett et al. 2016;Sachs et al. 2016;Frey et al. 2017) (Fig. 6b, c, e, g). Further, it is possible that the femur and humerus have approximately the same shape or they may differ (Hawkins 1840;Großmann 2006;Schumacher and Martin 2015;Sachs et al. 2016) (e.g., Fig. 6a, e).
Radius, ulna, tibia, and fibula are shortened and may diverge from the hourglass long-bone shape they have in basal Eosauropterygia (Rieppel 2000) becoming more rounded and disc-like (Andrews 1910;O'Keefe 2002;Großmann 2006;Sato et al. 2006;Druckenmiller and Russell 2008;Schumacher and Martin 2015;Sachs et al. 2016;Frey et al. 2017) (Fig. 5b, c). Often, accessory ossicles are present at the level of the zeugopodium or carpus/tarsus on the side of humerus or femur which is flared (Andrews 1910;Sato and Storrs 2000;Großmann 2006;Sato et al. 2006;Smith 2007;Druckenmiller and Russell 2008;Schumacher and Martin 2015;Sachs et al. 2016;Frey et al. 2017) (Fig. 6a-g). The fifth metacarpal and metatarsal have moved into the row of distal carpals/tarsals (Robinson 1975). The digits show hyperphalangy. Joint surfaces of the successive digits do not always lie in the same plane. The middle digit is often the longest. They become successively shorter in digit II and IV and then in I and V (Caldwell 1997, compare to e.g., Hawkins 1840Andrews 1910;Großmann 2006;Druckenmiller and Russell 2008;Schumacher and Martin 2015;Delsett et al. 2016;Sachs et al. 2016;Frey et al. 2017). All four plesiosaur limbs have evolved into hydrofoil-like flippers (Robinson 1975(Robinson , 1977Wintrich et al. 2017a). The flipper profiles were asymmetrical (Robinson 1975;Caldwell 1997;Frey et al. 2017) as those of recent underwater fliers, i.e., sea turtles and penguins (Fish 2004).
Joints-The proximal articular bony surfaces of humeri and femora of Cryptoclidus eurymerus are very incongruent with the bony glenoid and acetabular articulation surfaces. The glenoid is formed by scapula and coracoid and is oval in shape. Its long axis lies approximately in horizontal direction (Fig. 7a). The acetabulum is formed by all three hip bones (Druckenmiller and Russell 2008). It is oval in shape, but its pre-axially to post-axially axis is longer and its dorsoventral axis is shorter than in the glenoid.   (2015). f Shows pre-axial and g postaxial accessory ossicle formation. The posterodistal humeral (c) and femoral (f) articulation surfaces imply that maybe a lost or poorly ossified/cartilaginous accessory ossicle was present in C. eurymerus comparably to the more proximal one of the two accessory ossicles in P. latipinnis (g). Not to scale. ao accessory ossicle, ? unknown bony element 2018) and show a roughened surface (personal observation) (Fig. 4g, h). The only extant species for which a comparably structured proximal humeral articulation surface is known is Dermochelys coriacea. In Dermochelys coriacea (Rhodin et al. 1981: 245, fig. 1c andSnover andRhodin 2008: 24, fig. 2.3) the humeral articulation surface is uniquely covered by a thick, vascularized cartilage cap which is non-parallel to the underlying bone (compare to Rhodin et al. 1981: 245, fig. 1b andSnover andRhodin 2008: 24-25, fig. 2.5-2.6). This is opposed to the hyaline cartilage capping of articular surfaces of other turtles and tetrapods. Hyaline cartilage covers markedly smoother bone surfaces. It lacks vascularization and is nourished by the synovial fluid that fills the joint capsule by diffusion. Thus, hyaline cartilage is only a few millimetres in thickness and, in comparison to the thick vascularized cartilage of Dermochelys coriacea, it is relatively parallel to the underlying bone surface (Rhodin et al. 1981;Snover and Rhodin 2008). Therefore, in comparison to Dermochelys coriacea, it is possible that the proximal epiphyses of plesiosaur humeri and femora were similarly capped by thick, vascularized cartilage that is not parallel to the underlying bone surface. At mid-shaft, humerus and femur are oval in transverse cross section (Krahl et al. 2013;Wintrich et al. 2017a). The elbow and carpal joints are stiffened (Storrs 1993).  Andrews (1910), note the oval shape of the glenoid; b pelvic girdle of Cryptoclidus eurymerus, redrawn and mirrored from Andrews (1910), the acetabulum forms a long oval shape; c pectoral girdle and humerus of a Caretta caretta after dissecting off the muscles, glenoid shown with intact hyaline cartilage cover (results of muscle dissection published as Krahl et al. 2019); d rough sketch of the sea turtle pectoral girdle redrawn from Wyneken (2001) for orientation. Please note how similar the rough shape of the sea turtle glenoid (c, d) resembles the plesiosaur glenoid (a), both formed by glenoid and coracoid. Drawn not to scale

Mode of locomotion-Proposed modes of locomotion
Plesiosaurs are unique among recent and fossil tetrapods in that they evolved four very similarly shaped flippers. Plesiosaurs were long regarded as rowers or paddlers (Fig. 8a). In both cases, drag-based propulsion is used. By performing a rowing or paddling movement, the flipper is maximally spread and pushes against the water to propel the body forward. The recovery stroke is carried out with as little water resistance as possible (Williston 1914;Tarlo 1958). Recently, this hypothesis was reactivated again based on muscle reconstructions (Araújo and Correia 2015;). Rowing appears to be useful in complex habitats, in which a lot of manoeuvering, acceleration, and deceleration is necessary (Walker and Westneat 2000;Fish et al. 2003). Godfrey (1984), Lingham-Soliar (2000), and Liu et al. (2015) suggested a further hypothesis, i.e., that plesiosaurs employed the so-called "rowing-flight" of sea lions (Otariinae) (English 1976b;Feldkamp 1987) (Fig. 8b). This locomotory mode is a combination of underwater flight and rowing elements, i.e., a combination of lift-based and dragbased propulsion. Initially the sea lion draws its foreflippers downwards (ventrally), resulting in lift and propulsion. When the maximum ventral flipper excursion is reached, the sea lion pulls its foreflippers suddenly backwards in a lateral rowing motion. When the body sinks slightly downwards, the foreflippers are passively rotated into the initial position (English 1976b;Feldkamp 1987). Liu et al. (2015) computationally modelled plesiosaur locomotion and find support for a rowing-flight stroke under certain circumstances, i.e., if maximum flipper excursions are presumed.
A third swimming hypothesis was proposed by Robinson (1975Robinson ( , 1977, the underwater flight, which has found the greatest endorsement so far by Lingham-Soliar (2000) (2019). In contrast to rowing, in underwater flight the flipper is always moved through the water at an angle of rotation smaller than 90° (Davenport et al. 1984;Walker and Westneat 2002). Because of the arched flipper profile, a net overpressure results on the underside of the profile and an underpressure on the upper side of the profile and the flipper is "sucked" and "pushed" forward at the same time by the pressure gradient (Baudinette and Gill 1985;Fish 1996;Walker and Westneat 2000). This type of locomotion is well suitable for long-distance swimmers travelling at moderate speeds (Walker and Westneat 2000).
Requiring additional discussion is the so-called "four wing problem". This is based on the unique situation that plesiosaurs have four uniformly-shaped hydrofoil flippers (Halstead 1989). The question is how the four flippers were coordinated and how the hindflippers avoid the vortex wakes of the foreflippers. Frey and Riess (1982) and Tarsitano and Riess (1982) advocate that the fore-and hindflippers were moved alternatingly, thus a continuous propulsion is ensured. That way, the hindflippers are prevented from entering the vortex wakes of the foreflipper. The recovery stroke was then supposed to have occurred passively (Frey and  (1987); c "synchronous rowing" in the pig-nosed turtle Carettochelys insculpta, after Rivera et al. (2013); d underwater flight in the sea turtle Caretta caretta, after Rivera et al. (2013) Riess 1982; Tarsitano and Riess 1982;Muscutt et al. 2017). Lingham-Soliar (2000) and Carpenter et al. (2010) endorsed the alternative hypothesis that all four flippers were moved synchronously. The plesiosaur would have moved simultaneously up-and forward, thus avoiding the danger of plunging into the vortex wakes. Long et al. (2006) carried out experiments with a swimming robot and suggested that plesiosaurs had a repertoire of different gaits, asynchronous, synchronous, semi-synchronous. These were used when needed, and they might have varied between taxa (Long et al. 2006). Based on assumptions of a variety of fore-and hindflipper motion ranges, Liu et al. (2015) show with computer simulations that the three proposed gaits are possible (see also Muscutt et al. 2017). This does not appear unlikely, because plio-and plesiosauromorph plesiosaurs have different hunting and dietary strategies (see above).

Mode of locomotion-Discussion of different modes of locomotion
In particular the stiffness of the trunk analogous to chelonioids, the shortened tail, and the long necks of the plesiosauromorph plesiosaurs, imply the necessity of evolving a paraxial mode of swimming (Storrs 1993;Lingham-Soliar 2000) but which one was employed by plesiosaurs is still controversially discussed. The enlargement of the ventral pectoral and pelvic girdle elements anterior and posterior to glenoid and acetabulum (Watson 1924;Tarlo 1958;Godfrey 1984) accompanied by the reduction of the dorsal bony elements (scapular blade and ilium) were used as arguments for rowing or a rowing element in plesiosaur locomotion (Frey and Riess 1982;Tarsitano and Riess 1982;Godfrey 1984). This is because it seems to imply relative hypertrophy of especially the locomotor muscles used in protraction and retraction originating on the coracoid and the pubis (and to a minor extent on the ventral side of the scapula and the ischium) as the respective authors argue. At the same time, this would indicate a reduction of musculature dorsal to the glenoid/acetabulum that is necessary for humeral and femoral elevation (Frey and Riess 1982;Tarsitano and Riess 1982;Godfrey 1984;Araújo and Correia 2015). Godfrey (1984) argues that the pectoral girdle in recent underwater fliers is a strong dorsoventral support structure that suspends the forelimb to the rib cage and vertebral column and that hampers the limb's displacement during a flipper beat cycle. Contrastingly, in plesiosaurs neither the scapula nor the ilium is tightly connected to the vertebral column. Therefore, he concluded that recent penguins and sea turtles may be poor analogues for plesiosaurs. Instead, he introduced sea lions that employ rowing-flight as functional analogues (Godfrey 1984).
The concept of underwater flight in plesiosaurs (Fig. 8d) is based on the form of glenoid and acetabulum which allows mostly elevation and depression and restricts protraction and retraction (Storrs 1993). Robinson (1975) notes that plesiosaur flippers are hydrofoil-shaped, i.e., they converge distally in a pointed tip, are pre-axially thicker than post-axially, and have an arched flipper profile (Robinson 1975;Caldwell 1997). These are criteria which are not met by rowing tetrapods which have webbed and blunt-ended paddle-like feet (Robinson 1975). Sea lions and Carettochelys do not meet these characteristics either (English 1976a(English , 1977Rivera et al. 2013) (Fig. 5e). The hydrofoil-shape of plesiosaur flippers seems to be corroborated by the discovery of preserved flipper soft tissue (Dames 1895;Frey et al. 2017). Flipper hydrofoils are usually utilized in underwater flight, as in Chelonioidea (see below) and Spheniscidae (Robinson 1975(Robinson , 1977Lingham-Soliar 2000;Carpenter et al. 2010;Liu et al. 2015).

Mode of locomotion-Carettochelys mode of locomotion
A recent study by Rivera et al. (2013) shows that the pignosed turtle Carettochelys insculpta, a freshwater turtle, usually considered as convergent to Chelonioidea, also falls into the spectrum between rowing and underwater flight, like Otariinae. However, on the one hand, the flippers of Carettochelys move through a much larger dorsoventral excursion range than in "true" rowing turtles, but the range is significantly smaller than in Chelonioidea. On the other hand, the anteroposterior range of motion of the foreflipper is significantly greater than in sea turtles. Most of the flipper rotation needed for the rowing motion is provided by humerus rotation. Compared to sea lions, the underwater flight phase of Carettochelys occupies a much smaller percentage of the locomotory cycle (Rivera et al. 2013). Both Otariinae (Feldkamp 1987) and Carettochelys illustrate that the dichotomy of underwater flight vs. rowing might actually be considered as a continuum (Rivera et al. 2013). The flipper beat cycle of Carettochelys is discussed for the first time as a possible gait for plesiosaurs (Fig. 8c). This gait is not considered as the main swimming mode of plesiosaurs but as an option for e.g., changing direction or surfacing to breath. Further, this adds another paraxial swimming style observed in an extant taxon to the general discussion on paraxial locomotion of extinct marine Tetrapoda, that has been missing in this discussion so far.

Locomotory apparatus and locomotion of Chelonioidea
Osteology-The chelonioid trunk is stiffened by the carapace and plastron. The turtle shell is flattened and has a streamlined body outline which is hydrodynamically optimized for a life in water (Davenport et al. 1984;Dziomber et al. 2021). In aquatic (marine) and semi-aquatic (freshwater) turtles, the coracoid is expanded and enlarged and the scapula is reduced as compared to tortoises (Walker 1973;Wyneken 1997;Depecker et al. 2006). Thus, in freshwater and marine turtles, the dorsal musculature, which originates at the scapula, is significantly reduced, while the coracoid musculature is hypertrophied (Walker 1973). The humerus of Chelonioidea shows morphological specializations in comparison to tortoises and semi-aquatic turtles, such as a v-shaped deltopectoral crest and a greatly enlarged medial process (Hirayama 1992;Evers and Benson 2019) implying hypertrophy of the associated muscles (Walker 1973).
The humerus is dorsoventrally flattened, and its shaft is oval in cross section (Walker 1973;Zug et al. 1986;Hirayama 1994;Wyneken 2001) (Fig. 4e). The actual flipper is formed by radius, ulna, and the manus (Walker 1973), without a contribution from the humerus. The flipper is hydrofoil-like (Walker 1973) and has an asymmetrical profile (Fish 2004). The ulna is shorter than the radius in proximodistal direction. Radius and ulna are usually levelled, but in sea turtles, the radius has been moved relatively palmar to the ulna. Chelonioids have a large pisiform, intermedium, and ulnare. Contrastingly to them, the radiale is small (Walker 1973;Evers and Benson 2019) (Fig. 5d).
Chelonioid flippers evolved by extension of the metacarpals and phalanges of the three middle digits (Shaffer et al. 1997;Richardson and Chipman 2003) (Fig. 5d). In the course of ontogeny, radius and ulna are tightly linked by connective tissue. Likewise, the digits are increasingly encased by connective tissue. Additionally, the foreflipper is covered by tough and scaly skin. Overall, this leads to a secondarily stiffened semi-rigid hydrofoil-like flipper. Contrastingly, the small, round hindflippers are not particularly stiffened or elongated (Walker 1973;Wyneken 1997), but may provide additional propulsion and are otherwise used for manoeuvering (Walker 1973;Davenport et al. 1984;Wyneken 1997).

Joints
The glenoid fossa of chelonioids is oval. Its long axis is significantly inclined in anterodorsal-posteroventral direction (compare to Wyneken 2001: 51, fig. 99, bottom right; personal observation) (Fig. 7c-d). The likewise oval humeral head (Fig. 4i) fits relatively well into the glenoid, it only appears to be slightly larger than the cavity (Walker 1973;Zug et al. 1986;Wyneken 2001; personal observation), i.e., the contact surface of humerus and glenoid is smaller than the whole humeral head. The ellipsoid shoulder joint of sea turtles allows humeral depression and elevation to a greater degree than protraction and retraction and does not allow rotation (Walker 1971;Davenport et al. 1984;Pace et al. 2001;Rivera et al. 2011). The mobile elbow functions in extension and flexion (Rivera et al. 2011). Flipper rotation appears to take place by rotation of the carpus against radius and/or ulna. The digits have only limited mobility [personal observation of a Caretta caretta specimen frozen and later thawed for a myological flipper dissection published in Krahl et al. (2019)].

Mode of locomotion
In chelonioid underwater flight, the foreflippers beat cyclically, approximating the form of a narrow skewed "O" from anterodorsal to posteroventral (Davenport et al. 1984;Rivera et al. 2011Rivera et al. , 2013 (Fig. 8). The hypothesis of Walker (1971), who described the flipper beat cycle as an oblique figure eight was not confirmed by subsequent authors (Davenport et al. 1984;Rivera et al. 2011Rivera et al. , 2013. Walker's (1971) description greatly influenced the work of several authors working on plesiosaur locomotion (see, e.g., Robinson 1975).
At the beginning of the downstroke, chelonioids rotate the leading edge of the foreflipper anteroventrally by rotating the carpus and manus, while during the upstroke it is rotated anterodorsally (Walker 1971). The foreflippers are thereby moved through the water at an angle of 40°-70° from the horizontal (Davenport et al. 1984). Testudines that row pull the flippers horizontally through the water (Davenport et al. 1984;Pace et al. 2001;Rivera and Blob 2010;Rivera et al. 2013). While propulsion in underwater flight is generated by both, upstroke as well as downstroke, the downstroke of Chelonioidea is observed to be more powerful than the upstroke (Walker 1971;Davenport et al. 1984;Wyneken 1997). This is in contrast to penguin underwater flight, in which up-and downstroke are equally efficient (Clark and Bemis 1979).
Nothosaurs have a swimming tail which contributed to locomotion (Carroll and Gaskill 1985). In sea turtles the tail is much reduced, paralleled by the loss of the m. caudofemoralis portion originating from it Walker (1973). In plesiosaurs the tail is much reduced, too. Although a tail fin may be present (Smith 2013), the tail probably contributed mostly to streamlining (Frey et al. 2017).
The reduction of dorsal bones and the expansion of ventral bones of the nothosaur and plesiosaur pectoral and pelvic girdle (Godfrey 1984;Kuhn-Schnyder 1987) is paralleled by aquatic turtles (independent of whether they live in freshwater and row or paddle or in salt water and fly underwater) (Walker 1973;Wyneken 1997;Depecker et al. 2006). Therefore, this trend is independent of the locomotory mode employed, but instead probably connected with the reduction of the impact of gravitation on the body by buoyancy in water and due to a shift of adaptations for terrestrial locomotion to aquatic locomotion (Depecker et al. 2006). This means that the similar trend observed in Sauropterygia probably delineates convergently the transition from life on land to a life in water, but that it does not allow inferences on whether they row or fly underwater, with the knowledge of today.
Furthermore, the reduction of the dorsal bony elements and the expansion of the ventral bony elements in plesiosaurs appear to have been driven to an extreme in comparison to the state found in turtles. As pachypleurosaurs and plesiosaurs have been proven to be viviparous (Cheng et al. 2004;O'Keefe and Chiappe 2011), therefore by inference with the phylogenetic bracket, nothosaurs were giving birth to life young as well (Griebeler and Klein 2019). This indicates that Sauropterygia were completely independent from the terrestrial environment, unlike aquatic turtles, and were therefore completely freed from supporting their bodies on land. That could be the reason why sauropterygians, and especially plesiosaurs, were able to reduce the dorsal projections of the pectoral and pelvic girdle more extremely in comparison to turtles.
The variably post-axially expanded plesiosaur humeral and femoral morphology and the possibly associated accessory ossicle formation (e.g., Andrews 1910;Sato and Storrs 2000;Großmann 2006;Sato et al. 2006;Smith 2007;Druckenmiller and Russell 2008;Schumacher and Martin 2015;Sachs et al. 2016;Frey et al. 2017) are partially accounted for in phylogenetic analyses (e.g. Benson and Druckenmiller 2014) (Fig. 6a-g). Nonetheless, they would need thorough investigations on whether this represents solely a phylogenetic, a temporal, or an ecological signal, too. It should also be checked, if the different observed patterns eventually fit into the scheme of flipper geometries determined by measuring aspect ratios by O'Keefe (2001b).
The glenoid and acetabulum of plesiosaurs greatly resemble the ellipsoid glenoids of sea turtles. Although, their long axes are differently oriented (horizontally) in plesiosaurs (personal observation, Storrs 1993) than in Chelonioidea (anterodorsally-posteroventrally) (Walker 1973;Wyneken 2001, personal observation) (Fig. 7a-d). It remains speculative what plesiosaur humeral and femoral articulation surfaces with their respective cartilaginous caps looked like ( Fig. 4g-h) because they were probably, convergently to Dermochelys coriacea, (Rhodin et al. 1981;Snover and Rhodin 2008) covered by thick vascularized cartilage caps and their articulation surfaces were not parallel to the underlying bone. During a foreflipper beat cycle, the sea turtle humerus is depressed and slightly retracted during the downstroke and then it is elevated and slightly protracted during the upstroke. The humerus is not rotated along its long axis at any point of the flipper beat cycle (Walker 1971;Davenport et al. 1984;Pace et al. 2001;Rivera et al. 2011). Considering the similarity between the chelonioid glenoid and the plesiosaur glenoid and acetabulum this leads to the surprising conclusion that plesiosaurs were probably unable to rotate their humeri and femora as extremely (sometimes up to 90°) as suggested by authors discussing the style of locomotion for plesiosaurs (e.g., Watson 1924;Tarlo 1958;Robinson 1975;Godfrey 1984;Lingham-Soliar 2000;Carpenter et al. 2010;Liu et al. 2015). A humerus or femur rotation by up to 90° would presuppose some kind of a ball-and-socket joint that plesiosaurs certainly did not have. This leads to interesting implications: The plesiosaur glenoids/acetabula may have diverged to some degree from the sea turtle type and actually allowed some degree of humeral/femoral rotation. Alternatively, it may have been greatly underestimated and understudied how much flipper twisting along the flipper chord length (suggested briefly by Robinson 1975;Liu et al. 2015;Witzel et al. 2015) contributes to plesiosaur locomotion. A combination of both may actually have been possible. It is difficult to compare the nothosaur glenoid to either turtles or plesiosaurs because it has not been described in detail. Further, thorough investigations of joint surfaces and ranges of motion of sauropterygian limb joints could yield new insights into the evolution of paraxial swimming in Sauropterygia.
Sea turtles have a functional elbow and wrist or intracarpal joint. The former allows simple flexion and extension of the flipper (Rivera et al. 2011(Rivera et al. , 2013 while the latter allows flipper rotation. The wrist or intracarpal joint has not been studied in detail to my knowledge. Contrastingly, the distal sea turtle flipper is much stiffened (Walker 1973). Unlike to sea turtles, the nothosaur elbow is believed to be immobile (Kuhn-Schnyder 1987). The arrangement of radius, ulna, and the carpals of the nothosaur foreflipper suggest that lower arm rotation was possible in nothosaurs like in sea turtles. Further, the nothosaur carpometacarpal and the intraphalangeal joints were probably mobile as well (personal observation). In the fore-and hindflipper of plesiosaurs, only the glenoid and acetabulum were functional joints (Storrs 1993). This needs further research, because at least the carpometacarpal/tarsometatarsal and intraphalangeal joints appear to be slightly mobile to some degree [personal observation on Cryptoclidus eurymerus (STIPB R 324)]. In sea lions intracarpal and carpometacarpal joints look as if they tightly interlock, just like in plesiosaurs. Detailed studies of the sea lion foreflipper revealed, that there is considerable variability in actual joint mobility. The mobility in each joint might appear insignificant but adds up over several joints proximodistally (English 1976a, b). In plesiosaurs, interphalangeal joint surfaces that do not align preaxially to post-axially are usually interpreted as a flipper stiffening mechanism (Caldwell 1997). Not in sea turtles, but in some cetaceans comparable non-aligned digital joints can be found e.g., in the humpback whale (Cooper et al. 2007a(Cooper et al. : 1127 fig. 3e, l). Humpback whales have symmetrical, lift producing hydrofoils as control surfaces (Fish and Battle 1995). Their flippers bend considerably during manoeuvers and flapping (Edel and Winn 1978). Flipper bending is a well-known phenomenon across aerial and aquatic animals with flapping appendages (Lucas et al. 2014). The interdigital space appears to be larger in the less ossified cetacean foreflippers than in plesiosaur flippers (compare e.g. Cooper et al. 2007aCooper et al. : 1127, l with e.g., Frey et al. 2017: 111, fig. 11). Unlike to cetaceans, plesiosaur flippers display more extreme hyperphalangy. Cooper et al. (2007b) suggest that the hyperphalangic flippers of e.g., a plesiosaur could be bend more smoothly than a flipper with longer and fewer phalanges resulting in the same amount of bending. Cooper et al. (2007b) are unsure how to interpret this functional adaptation and suggested that the smoother bending curve of the flipper might be hydrodynamically advantageous in vortex shedding. I propose that the ability to smoothly extend and flex the digits in plesiosaurs indicates flipper bending and additionally flipper twisting in plesiosaurs. Further, if the glenoid and acetabulum restricted long axis rotation to a large extent, which is given by the similarity to the sea turtle glenoid, flipper twisting would be inevitably needed for locomotion. Therefore, the issue of mobility within plesiosaur flippers would need investigation, considering the existing morphological differences between taxa. Nevertheless, the mobility of the carpometacarpal and the intraphalangeal joints of sea turtles also remains understudied. In nothosaurs, this issue has remained basically untouched. Yet, foreflipper (in plesiosaurs also hindflipper) joint mobility in all three taxa would be crucial for our understanding of paraxial locomotion in secondary aquatic tetrapods.
This article also suggested the locomotor style of Carettochelys (Rivera et al. 2013) as a possible locomotory style for plesiosaurs. Many of the osteological characteristics arguing against rowing in plesiosaurs might also argue against this type of locomotion. Yet, the flipper tip excursion path seems to fit more adequately the plesiosaur hindflipper movements that Liu et al. (2015) inferred from computer simulations (compare Fig. 8c to Liu et al. 2015: 8, fig. 4a, b) than the flipper tip excursion path described by Feldkamp (1987) for sea lions. Many osteological clues lead to the conclusion that plesiosaurs were not adapted for rowing or a combination of rowing and underwater flying as the main mode of locomotion but instead were more suited for underwater flight. Underwater flight is employed by long-distance travellers (Walker and Westneat 2000). However, it remains a possibility that nothosaurs and plesiosaurs were able to modify their locomotory style to rowing [comparable to sea turtles (Wyneken 1997)] or a rowing and flight combination [the locomotory swimming style employed by Carettochelys (Rivera et al. 2013) or otariids (Feldkamp 1987)] for changes of direction or for surfacing for breathing like sea turtles (Wyneken 1997). A gait modification by employing a combination of underwater flight and rowing might be preferrable in plesiosaurs over rowing because of their potentially limited ability to rotate the humeri and femora along their length axes.

Conclusions
While nothosaurs partially relied on tail propulsion, plesiosaurs and sea turtles exclusively rely on paraxial locomotion. The reduction of the dorsal bracing system and simultaneous increase in coracoid size in nothosaurs and plesiosaurs, often used as argument pro rowing in the past, is convergent to the secondary adaptation of aquatic turtles. In turtles, this suit of adaptations reflects the change from terrestrial to aquatic limb-based propulsion and the diminishing of the impact of gravity on the body by buoyancy. However, this suit of adaptations in turtles does not inform on the aquatic mode of locomotion (i.e., rowing, a combination of rowing and flying, or flying) they employ. Nothosaurs and plesiosaurs are not comparable to sea turtles in that the former evolved their flippers by hyperphalangy and the latter by phalangeal elongation. Superficially, sea turtles and plesiosaurs are comparable because both have cambered hydrofoil flippers unlike nothosaurs that evolved more paddle-like flippers. A comparison of the plesiosaur glenoid and acetabulum to the glenoid of chelonioids shows, that plesiosaurs were probably unable to rotate their flippers as much as depicted in many reconstructions of swimming plesiosaurs so far. Flipper long axis rotation might have been possible but only in a largely restricted way. Studying the joints (glenoid, acetabulum, carpometacarpal, tarsometatarsal, and interphalangeal) of plesiosaurs and nothosaurs in comparison to recent taxa could shed light on degrees of freedom and actual motion ranges which would be necessary to paint a realistic picture of plesiosaur and nothosaur locomotion on an osteological basis, including flipper rotation and flipper twisting. The mode of locomotion employed by Carettochelys insculpta is discussed for the first time as an option for plesiosaur manoeuvering. Many osteological clues point to underwater flight as the main mode of locomotion for plesiosaurs. While plesiosaur locomotion has been subject to a number of studies, nothosaur locomotion has remained largely unstudied.
Funding Open Access funding enabled and organized by Projekt DEAL.
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