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Fish pectoral fins are roughly divided into large dorsal muscle mass, and a large ventral muscle mass. Just from that division and position of muscles, the ventral muscle mass seems to mainly lower the fin, flex it ventrally. And the dorsal muscle mass seems to flex it dorsally. While the compartments are termed "dorsal" and "ventral", the actual muscles are termed adductors and abductors. What is the rational for those terms?
The pectoral fin muscles of the coelacanth Latimeria chalumnae: Functional and evolutionary implications for the fin-to-limb transition and subsequent evolution of tetrapods
To investigate the morphology and evolutionary origin of muscles in vertebrate limbs, we conducted anatomical dissections, computed tomography and kinematic analyses on the pectoral fin of the African coelacanth, Latimeria chalumnae. We discovered nine antagonistic pairs of pronators and supinators that are anatomically and functionally distinct from the abductor and adductor superficiales and profundi. In particular, the first pronator and supinator pair represents mono- and biarticular muscles a portion of the muscle fibers is attached to ridges on the humerus and is separated into two monoarticular muscles, whereas, as a biarticular muscle, the main body is inserted into the radius by crossing two joints from the shoulder girdle. This pair, consisting of a pronator and supinator, constitutes a muscle arrangement equivalent to two human antagonistic pairs of monoarticular muscles and one antagonistic pair of biarticular muscles in the stylopod between the shoulder and elbow joints. Our recent kinesiological and biomechanical engineering studies on human limbs have demonstrated that two antagonistic pairs of monoarticular muscles and one antagonistic pair of biarticular muscles in the stylopod (1) coordinately control output force and force direction at the wrist and ankle and (2) achieve a contact task to carry out weight-bearing motion and maintain stable posture. Therefore, along with dissections of the pectoral fins in two lungfish species, Neoceratodus forsteri and Protopterus aethiopicus, we discuss the functional and evolutionary implications for the fin-to-limb transition and subsequent evolution of tetrapods. Anat Rec, 299:1203–1223, 2016. © 2016 Wiley Periodicals, Inc.
Developmental genetic, evolutionary, and paleontological studies of fins and limbs have established the well-advanced field of evolutionary developmental biology (Shubin and Alberch, 1986 Ahlberg and Milner, 1994 Johanson et al., 2007 Shubin et al., 2009 Boisvert et al., 2013 Tickle, 2015 ). Recently, research tools of molecular genetics and genomics, imaging analyses, discoveries and extensive studies of fossils and theoretical concepts all have brought, in particular, an integrated approach to investigate the fin-to-limb transition and have advanced our understanding of evolutionary developmental biology of fins and limbs (Johanson et al., 2007 Shubin et al., 2009 Clack, 2012 Amemiya et al., 2013 Pieretti et al., 2015 ). For instance, it has been known that endochondral skeletal elements of tetrapod limbs are arranged along the metapterygial axis (Shubin and Alberch, 1986 Mabee, 2000 ). Since the development of distal elements in limbs varies and is more elaborate in some sarcopterygian lineages (Cohn et al., 2002 Friedman et al., 2007 ), evolutionary and developmental mechanisms of the metapterygial axis have remained unclear. Onimaru et al. ( 2015 ) recently reported, however, that an alteration of anterior–posterior patterning of pectoral fins might have involved an evolutionary transformation of polybasal axes to the metapterygial axis in the fin-to-limb transition. In addition, recent evolutionary and developmental studies on the autopod have discovered regulatory elements in genomes of extant osteichthyan fishes that might have contributed to the morphological evolution of the autopod in the fin-to-limb transition (Amemiya et al., 2013 Schneider and Shubin, 2013 Gehrke et al. 2015 Pieretti et al., 2015 ).
Unfortunately, the majority of these studies have been skeletal-based and morphogenetic studies of the fin and limb musculature have seriously lagged, despite the fact that there has been information on comparative anatomy of the fin and limb musculature relevant for studies on the fin-to-limb transition (Humphry, 1872 Millot and Anthony, 1958 Winterbottom, 1973 Ahlberg, 1989 Ashley-Ross, 1995 Walthall et al., 2006 Diogo and Abdala, 2007 Diogo, 2008 Diogo et al., 2009 Abdala and Diogo, 2010 Diogo and Ziermann, 2015 Wilhelm et al., 2015 ). Since the muscles and skeleton develop interactively and because the skeleton does not form properly in the absence of tendon, ligament, and muscle development (Kardon, 1998 Kahn et al., 2009 ), evolutionary, developmental, and genetic studies of the musculature are critically needed (Chen, 1935 Kardon, 1998 Cole et al., 2011 ).
Antagonistic pairs of mono- and biarticular muscles are characteristic of human limbs (Kumamoto et al., 1994 ). However, the biology of these muscles has not been well discussed in the context of the development and evolution of other tetrapod limbs. In addition, the existence of biarticular muscles passing over two adjacent joints poses a paradox in biomechanics and functional biology. As pointed out by van Ingen Schenau ( 1989 ), when the rectus femoris acts in the thigh, the knee is extended but the hip is flexed. As the antagonistic muscles of the rectus femoris, when the semitendinosus, semimembranosus and biceps femoris long head act in the thigh, the knee is flexed but the hip is extended. When all the biarticular muscles in the thigh function, however, they act as a brake against each other in one joint, whereas they act coordinately in the other joint. To resolve this paradox and examine functional roles and the motion control of mono- and biarticular muscles, a theoretical link model for the human stylopod (upper arm or thigh) has been proposed and tested using electromyographic kinesiology, biomechanical engineering, and robotics (Kumamoto et al., 1994, 2000 Oshima et al., 2000 Fujikawa et al., 2001 Kanayama et al., 2001 Oshima et al., 2001 ). The proposed model consists of the coordinated motion control of the two-joint link mechanism equipped with two antagonistic pairs of monoarticular muscles and one antagonistic pair of biarticular muscles in the human stylopod. In this model, the requisite muscles, defined as the functionally different effective muscular system (FEMS), coordinately control the output force and force direction at the wrist or ankle and become indispensable for achieving weight-bearing motions and maintaining stable posture (Kumamoto et al., 1994 Kumamoto et al., 2000 Oshima et al., 2000 Fujikawa et al., 2001 Oshima et al., 2001 ).
We began investigating the morphology and evolutionary origins of mono- and biarticular muscles in fins and limbs by examining the pectoral fin of the African coelacanth, Latimeria chalumnae, an extant sarcopterygian fish with the pectoral fin skeleton along a metapterygial axis (Shubin and Alberch, 1986 Mabee, 2000 ). Our anatomical dissections and computed tomography (CT) and kinematic analyses of the pectoral fin of L. chalumnae revealed that the musculature of this fin is composed of abductor and adductor superficiales and profundi along with nine antagonistic pairs of pronators and supinators. The first antagonistic pronator and supinator pair represents a muscle arrangement equivalent to the human two-joint link mechanism in the stylopod. Since terrestrial locomotion is required to achieve a contact task for weight-bearing motions and stable posture on the ground, the coordinated motion control model of the two-joint link mechanism equipped with mono- and biarticular muscles may meet the minimum requirements for this achievement. Thus, our study on the musculature of the pectoral fin of L. chalumnae provides new functional and evolutionary implications for the fin-to-limb transition and subsequent evolution of tetrapods.
† These authors contributed equally to this study.
Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4695698.
Published by the Royal Society. All rights reserved.
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1. Material and methods
1.1. Species and specimens
We examined four species of anglerfish: two species of deep-sea pelagic swimmers and two species of benthic substrate locomotors (Figs 1, 2). Species and specimens were chosen primarily based on their locomotor ecology, completeness, and commonality in the Museum of Comparative Zoology (MCZ) Ichthyology Collection. Specimens were also chosen by their ability to be contrast-enhanced stained and μCT scanned in a reasonable timeframe, as the largest specimens of some anglerfish species can grow too large to efficiently stain or scan using a standard μCT scanner. The pelagic species include the triplewart seadevil, Cryptopsaras couesii (MCZ:Ich:76459 Fig. 2a), and the ghostly seadevil, Haplophryne mollis (MCZ:Ich:161516 Fig. 2b). Both are bathypelagic swimmers and are found globally in deep waters at depths as great as 2000m (Froese & Pauly Reference Froese and Pauly 2018). The two benthic species include the scarlet frogfish, Antennarius coccineus (MCZ:Ich:6807 Fig. 2c), and the oval batfish, Ogcocephalus notatus (MCZ:Ich:45075 Fig. 2d). The scarlet frogfish is a benthic reef-dwelling fish that reaches sizes of up to 91mm it is found largely in the tropical Indo-Pacific around Australia and in the waters of East Africa, the Red Sea, and California, at depths of up to 75m (Pietsch & Grobecker Reference Pietsch and Grobecker 1987). The oval batfish is also benthic but inhabits more open seabeds rather than reefs and can grow up to 130mm it is found in tropical waters of the West Atlantic and Caribbean Sea at depths of 15–172m (Froese & Pauly Reference Froese and Pauly 2018). Compared to the frogfishes, which have round bodies and are inefficient swimmers, batfishes are dorsoventrally compressed and can swim in short bursts. Both frogfish and batfish use their fins to ‘walk' around their habitat.
Figure 2 Full-body digital reconstructions with segmented pectoral girdles and fins overlaid. (A) Pelagic deep-sea swimming species Cryptopsaras couesii (triplewart seadevil) and Haplophryne mollis (ghostly seadevil). (B) Benthic substrate locomotor species Antennarius coccineus (scarlet frogfish) and Ogcocephalus notatus (oval batfish).
1.2. Tissue staining, μCT scanning, and segmentation
To visualise the 3D morphology of the muscles and skeleton of the pectoral fin, whole alcohol-preserved specimens were stained to enhance soft tissue contrast with 2.5 % w/v phosphomolybdic acid (PMA) in 70 % ethanol for up to 21 days. Of all the common contrast agents available, iodine (I2KI) being the most common, PMA provides the best discrimination of all soft tissues, including cartilage (Descamps et al. Reference Descamps, Sochacka, De Kegel, Van Loo, Van Hoorebeke and Adriaens 2014). In order to minimise overstaining, specimens were scanned at intervals during the staining processes to track penetration of the stain. All specimens were scanned in the MCZ using a Bruker SkyScan 1173 (for settings, see Table 1). Once tissues of the pectoral fin were fully penetrated by the contrast stain, the bones, cartilage, and individual muscles were digitally dissected by manual segmentation using Materialise Mimics ® , and identified following the text and diagrams of Monod ( Reference Monod 1960). The bones/areas of muscle attachment at the origin and insertion were also identified.
Table 1 Species, specimen numbers, μCT scan settings, and days immersed in contrast-enhanced stain. Abbreviations: MCZ = Museum of Comparative Zoology, Harvard University Al = aluminium.
1.3. Individual muscle properties
Using Materialise Mimics ® , two muscle properties were digitally measured on each muscle: muscle volume and fibre length. Muscle volume was recorded directly from each segmented mesh. Fibre lengths were determined by using the measure distance tool on three separate fibres for each muscle and taking the average. Physiological cross-sectional area (PCSA) was then calculated for each muscle as:
The specific tension of lophiiform muscle is currently unknown. Though data is available for other fish species, values range from 65kNm –2 in European eel (Ellerby et al. Reference Ellerby, Spierts and Altringham 2001) to 289kNm –2 in dogfish shark (Lou et al. Reference Lou, Curtin and Woledge 2002). As force is proportional to PCSA, we use PCSA as a proxy for force capacity. All muscle properties were normalised to body mass (M b) to allow comparison between species of different sizes. Using geometric similarity, volumes were divided by M b fibre lengths were divided by M b 1/3 and PCSAs were divided by M b 2/3 .
In addition to comparing muscle properties across the four species sampled, we generated a functional morphospace by plotting normalised PCSAs against normalised fibre lengths. As PCSA is an indicator of force and fibre length is proportional to the range of muscle shorting (or distance over which force can be generated), a functional morphospace provides a visualisation of the physiological trade-offs within and between muscles (Lieber Reference Lieber 2002).
1.4. Coordinate system
The species examined here all hold their pectoral fins in different orientations relative to the main horizontal axis of the body (Fig. 2). The pectoral fin of Cryptopsaras is held flat against the body with the ‘adductor' surface facing the body and is directed posteriorly. In Haplophryne the pectoral fin is also held flat against the body but is instead directed posterodorsally. The pectoral fin in Antennarius is directed posteriorly and has been pronated slightly such that the ‘adductor' surface faces more dorsally. Compared to Antennarius, the pectoral fin of Ogcocephalus is even further modified, with both the girdle being dorsoventrally compressed and the fin projecting more laterally from the body. The fin of Ogcocephalus is pronated even further than Antennarius such that the ‘adductor' surface faces dorsally.
To effectively compare different bones, joints, and muscle actions between pectoral fins that are oriented differently with respect to the body, we established a coordinate system. Here the glenoid is treated as a three-degree-of-freedom joint with the following actions: protraction/retraction is anterior/posterior movements along the horizontal plane of the body elevation/depression is dorsal/ventral movements around the horizontal plane of the body and pronation/supination is inward/outward rotation around the long axis of the fin. We also describe flexion/extension of the lepidotrichia as follows: flexion is movement of the lepidotrichia towards the ‘abductor' surface of the fin (or towards the substrate in benthic species) and extension is movement of the lepidotrichia towards the ‘adductor' surface of the fin (or away from the substrate in benthic species).
Fish muscle: the exceptional case of notothenioids
Fish skeletal muscle is an excellent model for studying muscle structure and function, since it has a very well-structured arrangement with different fiber types segregated in the axial and pectoral fin muscles. The morphological and physiological characteristics of the different muscle fiber types have been studied in several teleost species. In fish muscle, fiber number and size varies with the species considered, limiting fish maximum final length due to constraints in metabolites and oxygen diffusion. In this work, we analyze some special characteristics of the skeletal muscle of the suborder Notothenioidei. They experienced an impressive radiation inside Antarctic waters, a stable and cold environment that could account for some of their special characteristics. The number of muscle fibers is very low, 12,700–164,000, in comparison to 550,000–1,200,000 in Salmo salar of similar sizes. The size of the fibers is very large, reaching 600 μm in diameter, while for example Salmo salar of similar sizes have fibers of 220 μm maximum diameter. Evolutionary adjustment in cell cycle length for working at low temperature has been shown in Harpagifer antarcticus (111 h at 0°C), when compared to the closely related sub-Antarctic species Harpagifer bispinis (150 h at 5°C). Maximum muscle fiber number decreases towards the more derived notothenioids, a trend that is more related to phylogeny than to geographical distribution (and hence water temperature), with values as low as 3,600 in Harpagifer bispinis. Mitochondria volume density in slow muscles of notothenioids is very high (reaching 0.56) and since maximal rates of substrate oxidation by mitochondria is not enhanced, at least in demersal notothenioids, volume density is the only means of overcoming thermal constraints on oxidative capacity. In brief, some characteristics of the muscles of notothenioids have an apparent phylogenetic component while others seem to be adaptations to low temperature.
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Movement and function of the larval zebrafish’s pectoral fin
As with morphology, larval zebrafish demonstrate stage-specific fin movement and function. Larval zebrafish actuate their pectoral fins in coordination with the body axis during slow swimming. At the initiation of a bout of swimming, the fins commonly abduct together but quickly fall into an alternating pattern between left and right sides ( Fig. 3 ). Pectoral fins are also able to beat, in similar alternating bouts, without the participation of the body axis, although this is uncommon under normal conditions (Green et al. 2011). Pectoral fin movements by larvae, unlike those of adults, occur over a relatively narrow range of fin beat frequencies at Hz. During fast, axial swimming, the fins do not alternate but are held next to the body (Thorsen et al. 2004 Müller and van Leeuwen 2004 Green et al. 2011) with active adduction (Green and Hale 2012).
Typical movements of the pectoral fins and the body axis during slow swimming of larval zebrafish. Fins alternate between the left and right sides and in coordination with axial bending. Note bending of the fin during abduction (e.g., on right side at 40 ms), Scale bar, 1 mm. Reprinted with permission from Green et al. (2011).
Focusing on the movement pattern of an individual fin we see that, although the structure of the larval pectoral fin appears rather symmetrical, both superficially and in its musculoskeletal structure, its movement is strikingly asymmetric. During the abduction phase of the fin-beat cycle, the pectoral fin bends midway along its proximodistal axis so that the distal end curves backward ( Fig. 4 A). The abducting fin curves back at a consistent position % along the length of its proximal to distal axis at a consistent phase (Green et al. 2013). During the adduction phase of the fin-beat cycle, the fin remains nearly straight with no comparable local bending. Local bending could be produced by differences in motoneuron activity in the two phases of movement but neurophysiological recordings suggest that the difference in curvature of the fin is not actively controlled. No obvious difference in the activity pattern of individual motoneurons or in multiunit ventral root recordings suggest an active mechanism, either for generating fin bending during abduction or for stiffening during adduction and thus we hypothesize that the asymmetry is generated passively through the mechanics of tissues (Green and Hale 2012).
Fin bending and modeling of fluid movement associated with the pectoral fins. (A) Fin movement during abduction and adduction of the pectoral fins demonstrates asymmetry in bending between these phases of movement. (B) Flow modeled in a representation of fin morphology and movement of larval zebrafish. (C) Flow in a manipulated model in which the pectoral fin remains straight as they are abducted and adducted though the fin beat cycle. Fin bending during abductions increases fluid-folding in larval zebrafish, suggesting that it is adapted to support respiratory exchange (Green et al. 2013). Images are an output from modeling performed by M. H. Green and O. Curet in association with Green et al. (2013).
Potential roles of the pectoral fins in locomotion have been investigated, using experimental approaches (Green et al. 2011) and computational fluid dynamics (Green et al. 2013), and focusing on the behavioral context of slow, forward swimming. Morpholino injections that blocked translation of fgf24, a gene that participates in initiation of forelimb development (e.g., Fischer et al. 2003) prevented formation of the pectoral fins while leaving other aspects of morphology comparable to the typical larva (Green et al. 2011). The pattern of axial movement and the performance of slow swimming were not significantly different between morpholino-injected finless fish and normal fish ( Fig. 5 ). There was no difference in the stability of the body in either roll or yaw and the fish did not appear to compensate for loss of the fins with changes in axial movements. Thus, although coordinated with the axis, the pectoral fins of larvae do not appear to function in generating thrust or stabilizing the body during slow swimming.
Slow swimming by a finless larval zebrafish. A full bout of swimming at axial frequencies typical of slow swimming of larval zebrafish with fins. The magnified images of the head illustrate the stability of the finless fish in roll and yaw. Scale bar = 1 mm. Reproduced with permission from Green et al. (2011).
An alternative hypothesis for the function of pectoral fins in larvae is that the fin is a respiratory structure (Hunter 1972 Weihs 1980 Osse and van der Boogart 1999). For small organisms, like the larval zebrafish, the fluid boundary layer is thick compared with body size. This impacts fluid movement near the surface of the body. In larval zebrafish, the skin is the primary location for exchange of ions and gas, including oxygen exchange for respiration, and it was proposed that larvae may use the pectoral fins to exchange oxygen-depleted water near the body with distant oxygen-rich fluid. Dye-based flow visualization and computational fluid dynamics demonstrated that, indeed, the fins do pull fluid distant from the body toward the trunk and move fluid in the boundary layer away from the side of the body (Green et al. 2011, 2103). In this way they fold fluid along the body ( Fig. 4 B), a mechanism for chaotic mixing under viscous conditions (Ottino 1989 Strook et al. 2002 Ottino and Wiggins 2004).
Although features of larval pectoral fin morphology suggest the presence of a distinctive organization in the larval stage, it is difficult to address whether fin movement in larvae is adapted to function in gas and/or ion exchange. One feature of kinematics that suggests the fins are specialized for movement and mixing of fluid is the fin bending in abduction, observed experimentally and discussed above. To address whether that bending is important for exchange at the skin, computational modeling was used to compare effectiveness of a normally bending and a straight fin in moving fluid along the side of the body ( Fig. 4 C). Bending of the fin during abduction improved movement of fluid near the body when compared with the straight fin, suggesting that its movement is adapted for the function of the fin in respiration.
Examining behavior of the pectoral fins in response to different, or differently perceived, oxygen levels provides ways to test the hypothesis of respiratory function. Green et al. (2011) explored whether this movement of the fin was associated with respiratory function by examining behavior of the fins and axis under different dissolved oxygen conditions. Decreasing the level of dissolved oxygen in the water resulted in a significant increase in the number of bouts of fin movement expressed during a given period of time. Changes were specifically observed in fin-only behavior, as opposed to axial swimming or coordinated fin and axial movement. van Rooijen et al. (2009) found a similar result but with a different approach. They identified lines of zebrafish with mutations in the vhl gene, which is part of the hypoxia response pathway. These animals perceive typical environmental conditions as hypoxic. The larvae of vhl mutants gulp water and beat their fins continuously, consistent with a behavioral response to hypoxia response behavior under the proposal that movement of the fins functions to generate respiratory flow.
Together these complementary data on morphology, movement, physiology, and mechanics argue that the fins are adapted to serve as respiratory structures during larval development. They support the idea that the pectoral fin goes through a distinct stage of musculoskeletal morphology in the first few weeks post-fertilization, during which time the fins generate bending and overall movement that is appropriate for respiratory function. Following comparative data of Grandel and Schulte-Merker (1998) that show commonalities in morphology among taxa, I suggest that this respiratory function of the pectoral fins will be common to many small fish larvae.
Human Appendicular Skeleton
The appendicular skeleton is composed of the bones of the upper limbs (which function to grasp and manipulate objects) and the lower limbs (which permit locomotion). It also includes the pectoral girdle, or shoulder girdle, that attaches the upper limbs to the body, and the pelvic girdle that attaches the lower limbs to the body (Figure (PageIndex<9>)).
Figure (PageIndex<9>): The appendicular skeleton is composed of the bones of the pectoral limbs (arm, forearm, hand), the pelvic limbs (thigh, leg, foot), the pectoral girdle, and the pelvic girdle. (credit: modification of work by Mariana Ruiz Villareal)
The Pectoral Girdle
The pectoral girdle bones provide the points of attachment of the upper limbs to the axial skeleton. The human pectoral girdle consists of the clavicle (or collarbone) in the anterior, and the scapula (or shoulder blades) in the posterior (Figure (PageIndex<10>)).
Figure (PageIndex<10>): (a) The pectoral girdle in primates consists of the clavicles and scapulae. (b) The posterior view reveals the spine of the scapula to which muscle attaches.
The clavicles are S-shaped bones that position the arms on the body. The clavicles lie horizontally across the front of the thorax (chest) just above the first rib. These bones are fairly fragile and are susceptible to fractures. For example, a fall with the arms outstretched causes the force to be transmitted to the clavicles, which can break if the force is excessive. The clavicle articulates with the sternum and the scapula.
The scapulae are flat, triangular bones that are located at the back of the pectoral girdle. They support the muscles crossing the shoulder joint. A ridge, called the spine, runs across the back of the scapula and can easily be felt through the skin (Figure (PageIndex<10>)). The spine of the scapula is a good example of a bony protrusion that facilitates a broad area of attachment for muscles to bone.
The Upper Limb
The upper limb contains 30 bones in three regions: the arm (shoulder to elbow), the forearm (ulna and radius), and the wrist and hand (Figure (PageIndex<11>)).
Figure (PageIndex<11>): The upper limb consists of the humerus of the upper arm, the radius and ulna of the forearm, eight bones of the carpus, five bones of the metacarpus, and 14 bones of the phalanges.
An articulation is any place at which two bones are joined. The humerus is the largest and longest bone of the upper limb and the only bone of the arm. It articulates with the scapula at the shoulder and with the forearm at the elbow. The forearm extends from the elbow to the wrist and consists of two bones: the ulna and the radius. The radius is located along the lateral (thumb) side of the forearm and articulates with the humerus at the elbow. The ulna is located on the medial aspect (pinky-finger side) of the forearm. It is longer than the radius. The ulna articulates with the humerus at the elbow. The radius and ulna also articulate with the carpal bones and with each other, which in vertebrates enables a variable degree of rotation of the carpus with respect to the long axis of the limb. The hand includes the eight bones of the carpus (wrist), the five bones of the metacarpus (palm), and the 14 bones of the phalanges (digits). Each digit consists of three phalanges, except for the thumb, when present, which has only two.
The Pelvic Girdle
The pelvic girdle attaches to the lower limbs of the axial skeleton. Because it is responsible for bearing the weight of the body and for locomotion, the pelvic girdle is securely attached to the axial skeleton by strong ligaments. It also has deep sockets with robust ligaments to securely attach the femur to the body. The pelvic girdle is further strengthened by two large hip bones. In adults, the hip bones, or coxal bones , are formed by the fusion of three pairs of bones: the ilium, ischium, and pubis. The pelvis joins together in the anterior of the body at a joint called the pubic symphysis and with the bones of the sacrum at the posterior of the body.
The female pelvis is slightly different from the male pelvis. Over generations of evolution, females with a wider pubic angle and larger diameter pelvic canal reproduced more successfully. Therefore, their offspring also had pelvic anatomy that enabled successful childbirth (Figure (PageIndex<12>)).
Figure (PageIndex<12>): To adapt to reproductive fitness, the (a) female pelvis is lighter, wider, shallower, and has a broader angle between the pubic bones than (b) the male pelvis.
The Lower Limb
The lower limb consists of the thigh, the leg, and the foot. The bones of the lower limb are the femur (thigh bone), patella (kneecap), tibia and fibula (bones of the leg), tarsals (bones of the ankle), and metatarsals and phalanges (bones of the foot) (Figure (PageIndex<13>)). The bones of the lower limbs are thicker and stronger than the bones of the upper limbs because of the need to support the entire weight of the body and the resulting forces from locomotion. In addition to evolutionary fitness, the bones of an individual will respond to forces exerted upon them.
Figure (PageIndex<13>): The lower limb consists of the thigh (femur), kneecap (patella), leg (tibia and fibula), ankle (tarsals), and foot (metatarsals and phalanges) bones.
The femur , or thighbone, is the longest, heaviest, and strongest bone in the body. The femur and pelvis form the hip joint at the proximal end. At the distal end, the femur, tibia, and patella form the knee joint. The patella , or kneecap, is a triangular bone that lies anterior to the knee joint. The patella is embedded in the tendon of the femoral extensors (quadriceps). It improves knee extension by reducing friction. The tibia , or shinbone, is a large bone of the leg that is located directly below the knee. The tibia articulates with the femur at its proximal end, with the fibula and the tarsal bones at its distal end. It is the second largest bone in the human body and is responsible for transmitting the weight of the body from the femur to the foot. The fibula , or calf bone, parallels and articulates with the tibia. It does not articulate with the femur and does not bear weight. The fibula acts as a site for muscle attachment and forms the lateral part of the ankle joint.
The tarsals are the seven bones of the ankle. The ankle transmits the weight of the body from the tibia and the fibula to the foot. The metatarsals are the five bones of the foot. The phalanges are the 14 bones of the toes. Each toe consists of three phalanges, except for the big toe that has only two (Figure (PageIndex<14>)). Variations exist in other species for example, the horse&rsquos metacarpals and metatarsals are oriented vertically and do not make contact with the substrate.
Figure (PageIndex<14>): This drawing shows the bones of the human foot and ankle, including the metatarsals and the phalanges.
Evolution of Body Design for Locomotion on Land
The transition of vertebrates onto land required a number of changes in body design, as movement on land presents a number of challenges for animals that are adapted to movement in water. The buoyancy of water provides a certain amount of lift, and a common form of movement by fish is lateral undulations of the entire body. This back and forth movement pushes the body against the water, creating forward movement. In most fish, the muscles of paired fins attach to girdles within the body, allowing for some control of locomotion. As certain fish began moving onto land, they retained their lateral undulation form of locomotion (anguilliform). However, instead of pushing against water, their fins or flippers became points of contact with the ground, around which they rotated their bodies.
The effect of gravity and the lack of buoyancy on land meant that body weight was suspended on the limbs, leading to increased strengthening and ossification of the limbs. The effect of gravity also required changes to the axial skeleton. Lateral undulations of land animal vertebral columns cause torsional strain. A firmer, more ossified vertebral column became common in terrestrial tetrapods because it reduces strain while providing the strength needed to support the body&rsquos weight. In later tetrapods, the vertebrae began allowing for vertical motion rather than lateral flexion. Another change in the axial skeleton was the loss of a direct attachment between the pectoral girdle and the head. This reduced the jarring to the head caused by the impact of the limbs on the ground. The vertebrae of the neck also evolved to allow movement of the head independently of the body.
The appendicular skeleton of land animals is also different from aquatic animals. The shoulders attach to the pectoral girdle through muscles and connective tissue, thus reducing the jarring of the skull. Because of a lateral undulating vertebral column, in early tetrapods, the limbs were splayed out to the side and movement occurred by performing &ldquopush-ups.&rdquo The vertebrae of these animals had to move side-to-side in a similar manner to fish and reptiles. This type of motion requires large muscles to move the limbs toward the midline it was almost like walking while doing push-ups, and it is not an efficient use of energy. Later tetrapods have their limbs placed under their bodies, so that each stride requires less force to move forward. This resulted in decreased adductor muscle size and an increased range of motion of the scapulae. This also restricts movement primarily to one plane, creating forward motion rather than moving the limbs upward as well as forward. The femur and humerus were also rotated, so that the ends of the limbs and digits were pointed forward, in the direction of motion, rather than out to the side. By placement underneath the body, limbs can swing forward like a pendulum to produce a stride that is more efficient for moving over land.
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Materials and Methods
All experiments were carried out in accordance with relevant regulatory standards for animal ethics, and were approved by the University of Queensland Animal Welfare Unit.
Zebrafish (Danio rerio) of the Tupfel long fin (TL) strain were housed at 26ଌ, and fed a standard diet of live artemia. Larvae were raised in E3 media at 28.5ଌ, with regular water changes and cleaning. The larvae used in this study were tested at 7 or 14 dpf, with average lengths of 4.65 mm and 6.93 mm, respectively. Animals tested at 14 dpf were fed rotifers from 5 dpf until the time of testing.
Startle and prey capture
For startle experiments, larvae were placed, 2 or 3 per dish, into 55 mm Petrie dishes in E3 media. Dishes were placed into a custom-built Plexiglas platform with a 70 mm computer-controlled speaker mounted on it. This platform was placed under a Nikon SMZ-745T dissecting microscope and illuminated from below by the microscope's halogen lamp. The startle stimulus took the form of a 550 hz tone that was applied while high-speed imaging was in progress. The categorization of startles as being fin-active or fin-inactive was done by a scorer who was blind to the latency of the startle. For prey capture, 2 or 3 larvae were placed in a 55 mm Petrie dish with roughly 100 paramecia (Paramecium caudatum, Southern Biological, Nunawading, Victoria, Australia) and filmed as for startle behavior.
Movies were taken at 506 frames per second at a resolution of 1200 pixels using a Fastec HiSpec camera (Fastec Imaging, San Diego, CA, USA) mounted on a Nikon SMZ-745T dissecting microscope. Imaging began at the time of stimulus presentation for startle behavior. For prey capture, an operator triggered Fastec software upon observing a strike movement, thus saving the prior five seconds of data. Kinematics were quantified for each third frame in most cases, but less often during periods of inactivity.
Method for quantifying fin-tail coordination
For each frame analyzed, points were manually placed at (1) the anterior-most point, (2) the midline between the base of the pectoral fins, (3) the midline between the resting points for the tips of the pectoral fins, (4) the midline halfway from the pectoral fins to the tip of the tail, and (5) the tip of the tail. These five points laid out the midline of the animal, defined the animal's bearing (from point 2 to point 1), the tail bend (sum of the angles at points 3 and 4), and provided a baseline (from point 2 to point 3) from which to calculate pectoral fin extensions. Additional points were placed at the (6) base and (7) tip of the right pectoral fin, and the (8) base and (9) tip of the left pectoral fin. These allowed the angles of the fins to be compared to the baseline angle, yielding the degree of abduction of each of the fins. Finally, for prey capture events, a point (10) was placed on the prey. This permitted the bearing and distance to the prey to be calculated.
Coordinate data were exported into Microsoft Excel, where calculations were carried out as follows. Bearing was the angle of the line from the trunk midline to the tip of the nose. A bearing of zero was defined as the bearing at t =𠂠, and was positive if the animal rotated clockwise and negative if counterclockwise. For prey capture, bearing was the angle between the larva's bearing and the true bearing to the prey, and was positive if the prey was to the larva's right, negative if to the left. Tail bend was defined as the sum of the angles formed at the two most posterior angles of the schematic larva (positive for a right bend, negative for a left bend). Fin extension was the absolute value of the difference between the angle of the fin and the angle of the trunk where the fin would normally lie flat. Macros for ImageJ and templates for Excel will be provided upon request.
To quantify strike velocity, the distance travelled by the larva during seven frames of the high-speed movie spanning the capture were divided by 14 ms, the period of elapsed time during these frames. Total fin abduction following a strike was quantified by averaging the left and right maximal fin abductions for each fin movement, and then summing the averages if more than one fin movement followed the strike.
A table (Table S1) of mean values for the data presented in this paper is available as supplementary material.
Evaluating optimization rules with a generative model of fin-body coordination
Why do developing larvae change how they divide labor between the fins and body? In other words, what cost function are larvae optimizing when they regulate fin-body coordination? To address this question, we built a simple computational model that allowed us to parameterize the division of labor between the fins and body (Appendix 1—figure 1A, details in Materials and methods). In this control-theoretic model, a larva swam towards a target (up or down) by comparing the target’s direction to the direction it would swim without steering (its current posture). From this difference the larva generated a steering command. The larva steered its swim bouts by using its fins to generate lift and its body to direct thrust.
A one-parameter control system captures fin-body coordination in silico.
(A) Circuit diagram to transform pitch-axis steering commands into climbing swims using the body and pectoral fins. Steering commands are defined by the direction of a target in egocentric coordinates. The relative weight of commands to rotate the body (to direct thrust) and produce an attack angle with the fins (by generating lift) is dictated by fin bias ( α ). To model physical transformations from commands into kinematic variables, commands to the body and fins are filtered to impose empirically-derived ceilings and floors on posture changes and attack angles (see Materials and methods). Swim trajectory is defined by posture (fish propel where they point) but modified by attack angle and error ( ε ). (B) Empirical fin bias ( α ^ ), computed from maximal sigmoid slope (slope/(1+slope)), as a function of age with 95% confidence intervals. (C) Attack angle as a function of posture change, plotted as means of equally-sized bins. Climbs to 100,000 targets were simulated using empirical fin bias ( α ^ ) from 1, 2, and 3 wpf larvae, and at α = 0 for comparison. (D) Mean attack angle for simulated larvae with parameterized fin bias (line), superimposed on empirical attack angles and fin biases ( α ^ ) for each clutch at each age. Simulated attack angles at α ^ account for 79% of variation in empirically observed attack angles (R 2 ).
Simulated larvae coordinated their fins and bodies by controlling both effectors with a mutual command. To vary the ratio of fin and body actions (attack angles and posture changes, respectively), the command was differentially scaled for the fins and body. Commands to the fins were weighted by a fin bias parameter ( 0 ≤ α ≤ 1 ) and commands to the body by ( 1 - α ) . Effector-specific commands were therefore positively correlated (when α ≠ 0 and α ≠ 1 ) in a ratio equal to α / ( 1 - α ) . Given this formulation, we could infer empirical fin biases ( α ^ ) from the ratio of empirical attack angles and posture changes, given by sigmoid slope ( α ^ = slope/(1+slope) Equation 3). Empirical fin bias increased significantly with age (from 0.74 at 1 wpf to 0.92 at 3 wpf) like sigmoid slope, but ranged from 0 to 1 (Appendix 1—figure 1B Table 1). Commands were transformed into kinematic variables according to physical transfer functions (Appendix 1—figure 1A) that increased approximately linearly near the origin, such that weak commands were faithfully transformed to movement for large positive and negative commands, transfer functions reached asymptotes to model physical limitations. The asymptotes imposed empirically-derived constraints on the range of posture changes (−17.0 to +13.2°) and attack angles (−2.9 to +14.0°) of each bout. Additionally, Gaussian noise was added to swim trajectory to model errors in motor control and effects of external forces like convective water currents that move larvae ( ε ).
The model permitted simulation of larvae across development, because sigmoid slope (and therefore fin bias) captured developmental changes to swimming. We simulated larvae from each age group (100,000) identically, save for age-specific α ^ , as they climbed in series of bouts until reaching targets positioned half the tank away (25 mm). We placed the targets in directions randomly drawn from observed climbing trajectories (see Materials and methods). Simulated attack angles and posture changes were sigmoidally related, with steeper sigmoid slopes for older larvae (Appendix 1—figure 1C). Simply by varying fin bias, simulated larvae exhibited mean attack angles comparable to empirical values (Appendix 1—figure 1D). Simulated attack angles at age- and clutch-specific α ^ yielded close approximations of attack angle (R 2 = 0.79). A model with a single parameter that scales divergent commands can therefore produce fin-body coordination and mimic climbing behavior across development.
Increasing fin bias improves balance but costs effort in silico
Body-mediated climbing requires orienting the body upwards, causing posture to deviate from horizontal. Orienting upwards requires an initial energetic investment, but once larvae acquire the proper trajectory they can cease further turning. By comparison, fin-mediated climbing causes no postural deviation from horizontal and requires continued investment throughout the climb. Because larvae combine body- and fin-mediated climbing according to fin bias, we measured the effect of fin bias on posture variation and climbing efficacy.
We parameterized fin bias in silico, simulating larvae that climbed solely by generating lift with their fins ( α = 1 ) or solely by changing body posture ( α = 0 ), the former yielding larvae that never deviated from horizontal (Appendix 1—figure 2A). As fin bias increased, larvae remained closer to horizontal while climbing. After five bouts towards the steepest drawn target (63°), larvae swimming without using their fins ( α = 0 ) deviated 52° from horizontal, larvae with small fin bias (like those at 1 wpf, α ^ = 0.74 ) deviated 39°, and larvae with large fin bias (like those at 3 wpf, α ^ = 0.92 ) deviated only 13°.
Effects of fin-body coordination on balance-effort trade-off.
(A) Trajectories (lines) and initial positions (dots) of bouts simulated with the control system in (A) at fin biases of 0, 0.74 ( α ^ at 1 wpf), 0.92 ( α ^ at 3 wpf), and 1.0, for 1000 larvae swimming towards targets 25 μm away. Posture following the fifth bout of the steepest climb is superimposed. Scale bar equals 1 mm. (B) Simulated absolute deviation from horizontal posture as a function of α , plotted as mean (green line) and bootstrapped 99% confidence intervals (shaded band). Data are superimposed on empirical values for individual clutches of a given age (circles, Development, R 2 = 0.48) and otog-/- larvae (diamond). (C) Effort, the sum of squared motor commands to the body and fins, from simulations in (B) normalized and plotted as a function of α as mean (line) and bootstrapped 99% confidence intervals (shaded band). Empirical fin biases at 1, 2, and 3 wpf and for otog-/- larvae are indicated with triangles. (D) Cost as a function of fin bias, computed as sums of normalized curves in (B) and (C) weighted by β (balance weight) and ( 1 - β ), respectively (left). When β = 1 (green), cost is equivalent to normalized deviation from horizontal. When β = 0 (ochre), cost is equivalent to effort. Intermediate cost functions are plotted for β increasing by 0.2, with 99% confidence interval (shaded band). (E) Fin bias at which cost was minimized is plotted at each value of balance weight, with 95% confidence intervals. (F) Inferred balance weight ( β ^ ) is plotted as a function of age, with 95% confidence intervals. This weight gives the cost function minimized by empirical fin bias at a given age (from the curve in E).
We found that the relationship between posture variation and fin bias was similar for empirical and simulated larvae (Appendix 1—figure 2B). Larger fin biases were associated with smaller deviations from horizontal, reflecting better balance. Although the model was not explicitly fit to postural variables, simulated deviations from horizontal explained 48% of empirical variance for clutches and time-points across development (with fin biases spanning from 0.60 to 0.93). Additionally, at fin biases below 0.5, simulated deviations were consistent near 17°. otog-/- larvae exhibited very small fin bias ( α ^ = 0.06 ) and large deviations from horizontal similar to simulated values at low α (19.1°). We conclude that simulations accurately captured the consequences of fin bias for balance, with greater fin bias allowing larvae to remain nearer horizontal.
To quantify how fin bias influenced swimming efficacy when climbing to targets, we measured effort. In order to define effort, we made a simplifying assumption: that effort would scale monotonically with the size of the desired movement. To begin, we defined effort as the sum of squared motor commands for steering. Our initial choice to make effort scale quadratically with the control signal follows the convention in Todorov and Jordan (2002) Guigon et al. (2007).
Swimming with greater reliance on the fins, with larger fin bias, made climbing more effortful. Larvae at lower fin biases climbed farther in the same number of swims, benefiting from the cumulative effects of body rotation on posture (Appendix 1—figure 1A). Simulated larvae at 1 wpf gained two-thirds more elevation (4.25 mm) than larvae at three wpf (2.55 mm) and nearly three times as much as larvae swimming solely with the fins ( α = 1 , 1.54 mm).
Effort increased monotonically with fin bias (Appendix 1—figure 2C). Steering solely with the fins ( α = 1 ) required 32 times more effort, on average, than steering solely by rotating the body ( α = 0 ). Because larvae could ultimately achieve steeper trajectories when steering with the body, larvae often required more swims to reach a target at large fin biases. By simulating effort at empirical fin biases, we estimated that older larvae swam with greater effort efforts at empirical fin biases (relative to effort at α = 1 ) corresponded to 3.6%, 5.5%, and 7.9% at 1, 2, and 3 wpf, respectively (Appendix 1—figure 2C, triangles). Further, very small fin bias observed in otog-/- larvae approximately corresponded with the least effortful swimming (3.1% of effort at α = 1 ). Effort also increased monotonically as a function of fin bias when computed as the sum of absolute motor commands as well as squared or absolute accelerations (Appendix 1—figure 3). We conclude that larvae achieve the least effortful climbing at low fin biases. Furthermore, as larvae develop and adopt larger fin biases, they swim with increasing effort.
Steering cost functions computed from various formulations of effort.
(A) Cost as a function of fin bias for β (balance weights) of 0 (ochre, composed solely of effort), 0.2, 0.4, 0.6, 0.8, and 1 (green), for effort computed as the sum of absolute motor commands. (B) Inferred balance weight as a function of age, with bootstrapped 99% CI, for effort computed as the sum of absolute motor commands. (C) Cost as a function of fin bias for effort computed as the sum of squared accelerations. (D) Inferred balance weight as a function of age, with bootstrapped 99% CI, for effort computed as the sum of squared accelerations. (E) Cost as a function of fin bias for effort computed as the sum of absolute accelerations. (F) Inferred balance weight as a function of age, with bootstrapped 99% CI, for effort computed as the sum of absolute accelerations.
Given that low fin biases required less effort and large fin biases facilitated balance, we next related the consequences of fin bias in each domain. We composed cost functions from terms for both balance and effort (Appendix 1—figure 2D). Specifically, we tested whether combinations of balance and effort terms could prescribe specific fin biases for optimal swimming. Cost functions are inherently dimensionless, so we summed normalized curves for balance (deviation from horizontal as a function of α , Appendix 1—figure 2B) and effort (sum of square motor commands as a function of α , Appendix 1—figure 2C). To vary the relative importance of balance and effort terms, we weighted them by β (balance weight) and 1- β , respectively.
Our behavioral experiments established that, as they develop, larvae appear to coordinate their fins and bodies in such a way as to permit more balanced swimming (Figure 2). In our simulations, at each age larvae swam with a fin bias that minimized cost for a particular combination of balance and effort. Empirical fin biases minimized distinct cost functions composed from different balance weights (Appendix 1—figure 2E). From the cost functions that were minimized by empirical fin biases, we estimated the inferred balance weight ( β ^ ) at each age. Fin bias of larvae at 1 wpf minimized a cost function composed from a very low inferred balance weight (Appendix 1—figure 2F) β ^ = 0.12 ). Inferred balance weight increased by 2 wpf and significantly by 3 wpf, to 0.18 and 0.32, respectively. This framework is therefore consistent with our behavioral observations.
The specific values obtained for inferred balance weight reflect the choice to define ‘effort’ as the sum of the squared motor commands for steering. For comparison, we also computed effort as the sum of absolute motor commands, as well as the sum of squared or absolute accelerations (see Methods for simulating swimming). Inferred balance weight increased monotonically and significantly with age for all definitions of effort tested (Appendix 1—figures 3B, 3D, 3F). We parameterized β and found the optimal fin bias ( α * ( β ) ), the fin bias at which cost was minimized (Figure 2E, Appendix 1—figures 3A, 3C, 3E). As β increased and cost functions grew more similar to deviation from horizontal, cost was minimized at larger fin biases. When β = 0 and the cost function solely reflected effort, the optimal fin bias was that which minimized effort ( α * ( 0 ) = 0.24 ). Conversely, maximal fin bias was optimal for a range of cost functions that heavily weighted balance ( β > 0.72). Our model can therefore generalize for different formulations of ‘effort’.
By providing a framework to contextualize our observations, the model offers a way to understand the trade-offs facing developing larvae. We conclude that larvae regulate fin-body coordination to optimize balance and effort, and that development of fin-body coordination can be explained by an increase in the importance of balance relative to effort.
Conclusions about optimizing coordination
We modeled the development of coordination as an adaptive process driven by dynamic optimization rules (Kugler et al., 1980 Izawa et al., 2008). To explore how larvae might implement coordination optimization rules, we quantified the effects of coordination parameters on performance variables, balance and effort (Rigoux and Guigon, 2012 O'Sullivan et al., 2009). We defined balance as deviation of posture from the horizontal, per our behavioral observation that fin-body synergies could minimize deviation from horizontal posture (Figure 2A). As the precise form of ‘effort’ is unknown, we assumed only that it would increase monotonically. We estimated effort either by scaling the control variables, as previously postulated in Todorov and Jordan (2002) Guigon et al. (2007), or by scaling kinematic variables proportional to the forces produced. We observed that steering with the body had a distinct advantage: rotations reoriented the body toward the target, minimizing the need for subsequent steering. In our model the more a larva used its pectoral fins to climb, the more effort (however defined) it expended to reach its target. However, consistent with our behavioral observations, steering with fins enabled climbing without changing trunk posture. We conclude that optimal fin bias reflects the relative importance of balancing and minimizing effort.
Taking our behavioral data and simulations together, we conclude that body-dominated climbing of young larvae is optimized primarily for effort, while fin-body synergies of older larvae are optimized both for effort and balance. Postural stability emerges as a key performance variable with development, so we propose that balance plays a primal role in the development of coordinated locomotion.
Methods for simulating swimming
We made a generative swimming model in Matlab to estimate how fin bias impacts balance and effort while climbing. Simulated larvae moved in two dimensions (horizontal, x , and vertical, z ) by making series of swim bouts ( b = 1 , … , n ) of variable trajectory ( t ) and fixed displacement (1.27 mm, the mean empirical displacement across all ages). Larvae swam from an origin at (0,0) such that the position after bout b was determined by the trajectory of all preceding bouts. For the horizontal dimension, in mm:
Larvae swam until traversing ≥ 99% of both the horizontal and vertical distances from the origin to a target, located at distance d and angle ϕ from the origin, or ( d ⋅ cos ( ϕ ) , d ⋅ sin ( ϕ ) ).
Larvae could control t during each bout through body rotation ( r ( b ) ) and by creating an attack angle with the pectoral fins ( γ ( b ) ). Body rotations allowed larvae to control their posture, which defined the direction of thrust. Larvae began swimming at horizontal posture (0°), meaning Θ during bout b was given by the sum of rotations during that and all preceding bouts:
For each bout, trajectory was defined as the sum of posture ( Θ (b)), attack angle ( γ ), and a noise term ( ε , defined below):
To steer, larvae could directly vary γ with the fins or influence Θ by rotating their bodies.
Movement noise ( ε ) was introduced to model motor errors and convective water currents that push larvae while they swim. Assuming finless larvae actively produce no attack angles ( γ = 0 Figure 1B and C), their empirical attack angles reflect external forces ( ε = t - Θ ). Therefore, simulated ε for each bout was randomly drawn from a Gaussian distribution with a mean of 0 and standard deviation measured from empirical attack angles of finless larvae at 3 wpf (11.36°).
To make concerted posture changes and attack angles that steered toward a target, both r and γ were derived from a variable steering command ( c ( b ) , in degrees) that provided feedback about the direction of the target. This command was defined before each bout and gave the direction of the target before the bout in egocentric terms (relative to the posture, Θ ( b - 1 ) , and position of the larva ( x ( b - 1 ) , z ( b - 1 ) )). For a larva oriented toward the target, c = 0 such that no steering occurred. For the first bout, angle c equaled ϕ , and thereafter (for b > 1 )
Rather than swim upside-down, model larvae were assumed to make yaw-axis turns (side-to-side) to keep the target horizontally forwards if a larva swam past the target, its horizontal position was simply reflected about the horizontal position of the target, such that ( d ⋅ c o s ( ϕ ) - x ) was always greater than 0.
Commands for attack angle ( γ ′ ) and body rotation ( r ′ ) were computed as complementary fractions that summed to the common steering command, c . The relative magnitude of γ ′ and r ′ was dictated by fin bias, α (defined from [0,1]), according to
When α = 1 larvae steered solely by generating attack angles with the fins, and when α = 0 steered solely with posture changes. When α adopted intermediate values, the ratio of fin commands to body rotation commands was therefore α / ( 1 - α ) .
To transform commands ( γ ′ and r ′ ) into kinematic variables ( γ and r ), we modeled physical limitations as a ceiling and floor imposed with logistic functions. These physical transfer functions for the fins and body had maximal slopes of 1 and were constrained to the origin, faithfully transforming commands over a certain range but reaching asymptotes at positive and negative extremes (Appendix 1—figure 1A). The fin transfer function had asymptotes defined by empirical best-fit sigmoids to attack angle vs. posture change, averaged across ages (Table 1). The lower asymptote equaled γ 0 (−2.94°) and the upper asymptote equaled γ m a x + γ 0 (14.04°). Given that the fin transfer function was also constrained to have maximal slope of 1 and pass through the origin, attack angle for a given bout was computed from the fin command according to
where k = 0.24 deg −1 . The body rotation transfer function was also constrained to have maximal slope of 1, pass through the origin, and have a range defined by the middle 99.9% of empirical body rotations (from −16.98° to 13.15°). Body rotation for a given bout was computed from the rotation command according to
To assess correlations of γ ( b ) and r ( b ) at age-, phenotype-, and clutch-specific values of α ^ , we simulated 100,000 larvae at each fin bias swimming to one target at d =25 mm (half the length of the empirical tank). The direction of the target, ϕ , was randomly drawn from the positive lobe of a Gaussian distribution of mean 0 and standard deviation of 20.67° (that of trajectories of empirical bouts pooled across all ages). We also examined how deviation from horizontal, the mean of absolute value of simulated postures ( Θ ( b ) ), as well as mean attack angle varied as a function of α , parameterized from 0 to 1 in increments of 0.01. Given that simulated larvae could deviate widely from horizontal, we computed circular mean posture in Matlab using CircStat (Berens, 2009). After a simulated larva reached its target in n bouts, effort ( E ) was computed as the sum of squared steering commands,
For comparison, effort was also calculated as the sum of absolute motor commands ( Σ | c ( i ) | ), as well as functions of acceleration. Given that simulated larvae had constant bout duration and displacement, ignoring drag makes angular acceleration proportional to body rotation and dorsal acceleration proportional to the sine of attack angle. We normalized these kinematic variables to their maximum values and summed the two to compute effort equivalent to the sum of squared accelerations
and the sum of absolute accelerations
Bootstrapped confidence intervals were calculated by resampling simulated larvae 1000 times with replacement.
Cost function derivation
Cost ( Q ( α ) ) was calculated as a weighted sum of normalized deviation from horizontal ( Θ * ( α ) ), Appendix 1—figure 2B) and normalized effort ( E ( α ), Appendix 1—figure 2C), after both were interpolated fivefold and smoothed with a 25 point sliding window. Deviation from horizontal was scaled by a balance weight coefficient ( 0 ≤ β ≤ 1 ) and effort was scaled by (1- β ), such that
Parameterizing β yielded a family of cost functions. Finding the fin bias at which cost was minimized gave the optimal fin bias, α * ( β ) . Confidence intervals on optimal fin bias were taken as the farthest neighboring values of β , larger and smaller, at which the bootstrapped 2.5 percentile of cost exceeded the minimal cost. Inferred balance weights ( β ^ ), those weights giving cost functions minimized by empirical fin biases ( α * = α ^ ) empirical fin biases, were estimated by linear interpolation. Confidence estimates on β ^ were similarly interpolated from 95% confidence intervals of α * evaluated at 95% confidence intervals of α ^ .