Performance traits such as bite forces are crucial to fitness and relate to the niche and adaptation of species. However, for many insects it is not possible to directly measure bite forces because they are too small. Biomechanical models of bite forces are therefore relevant to test hypotheses of adaptation in insects and other small organisms. Although such models are based on classical mechanics, combining forces, material properties and laws of levers, it is currently unknown how various models relate to bite forces measured in vivo. One critical component of these models is the physiological cross-sectional area (PCSA) of muscles, which relates to the maximum amount of force they can produce. Here, using the grasshopper Schistocerca gregaria, we compare various ways to obtain PCSA values and use in vivo measurements of bite forces to validate the biomechanical models. We show that most approaches used to derive PCSA (dissection, 3D muscle convex hull volume, muscle attachment area) are consistent with the expected relationships between PCSA and bite force, as well as with the muscle stress values known for insects. The only exception to this are PCSA values estimated by direct 3D muscle volume computation, which could be explained by noisy variation produced by shrinkage. This method therefore produces PCSA values which are uncorrelated to in vivo bite forces. Furthermore, despite the fact that all other methods do not significantly differ from expectations, their derived PCSA values vary widely, suggesting a lack of comparability between studies relying on different methods.

Whole-organism performance (Lailvaux and Irschick, 2006) is the pivot between organismal morphology and fitness (Arnold, 1983). As such, it has become an indispensable component of studies of adaptation, notably in the related fields of evolutionary functional morphology and biomechanics (e.g. Herrel et al., 2010; Lailvaux, 2018; Santana et al., 2010). Individual performances, such as bite force, can either be modeled from morphological traits, or measured in vivo (Ginot et al., 2018; Gröning et al., 2013; Heethoff and Norton, 2009; Herrel et al., 2008). Reasons for preferring a modeling approach over in vivo measurements are numerous. (i) In vivo performance is subject to ‘noisy’ variation related to, for example, temperature or motivation of the measured individuals (Anderson et al., 2008). (ii) Researchers have fewer degrees of freedom to measure in vivo. For example, bite forces are easily measured at distal mandible parts, but less so at proximal locations (e.g. molars; Cox et al., 2012), where the strongest forces are used for breaking down the food (e.g. chewing). (iii) In vivo measurements simply cannot be accessed in some cases (e.g. extremely big or small organisms, collection specimens).

Historically, estimators of in vivo bite forces were based either on morphometric proxies such as mechanical advantage, areas of muscle insertion or muscle scars (e.g. Dickinson et al., 2021, and references therein), or on morphometric and muscular physiological data obtained from dissections [muscle mass and fiber length are used to compute physiological cross-sectional area (PCSA); e.g. Herrel et al., 2008; Martin et al., 2020; Powell et al., 1984]. The former methods represent proxies for muscle size or lever force transmission, and although they may be correlated to in vivo forces, they cannot always be used to actually estimate the forces themselves. Furthermore, their degree of correlation with muscle PCSA or force can vary widely (Dickinson et al., 2021; Ginot et al., 2019). The latter methods, when muscle stress values are known, can be used to compute estimates of forces, but require precise dissection work, and are limited, e.g. by the size of specimens. Therefore, dissection-based models have mainly been used for small- to medium-sized vertebrates (Genbrugge et al., 2011; Ginot et al., 2018; Hartstone–Rose et al., 2022; Herrel et al., 2008; Powell et al., 1984; Santana et al., 2010; Wittorski et al., 2016). In recent years, these dissection-based biomechanical models have tended to be replaced by 3D reconstruction-based approaches (e.g. Cox et al., 2012; Gröning et al., 2013; Gignac et al., 2016), which allow the reconstruction of hard and soft tissues, are usually less destructive, and have enabled smaller-sized organisms such as insects to be studied (Blanke et al., 2018; Martin et al., 2020; Püffel et al., 2021, 2022; Weihmann et al., 2015). Although methodological details differ, depending on the organism studied, these new methods mostly rely on chemical fixation and staining with X-ray contrast agents such as iodine to reveal soft tissues in subsequent microtomography (Gignac et al., 2016). In arthropods, it is also considered best practice to have critical point drying as an additional step before tomography (Mensa et al., 2022). It should be noted that the fixation and staining process is known to cause soft tissue shrinkage, regardless of the protocol used (Hedrick et al., 2018; Martin et al., 2020; Romeis, 1989; Vickerton et al., 2013). Because of this, the measurements obtained for fixed and stained specimens may not be directly comparable to measurements obtained from dissection of fresh or ethanol-preserved specimens. In addition, the different microtomographic devices and their settings can also impact 3D anatomical reconstructions, and should therefore be reported, although they may not be easily corrected for a posteriori (Gignac et al., 2016). In any event, these new methods aim to allow researchers to make inferences about forces, or more generally performance, and therefore any model or proxy should be validated against in vivo data (Arnold, 1983).

Bite force is a major whole-organism performance trait measured by biologists in the past 25 years (Anderson et al., 2008; Deeming et al., 2022; Gomes Rodrigues et al., 2022; Herrel et al., 1999; Isip et al., 2022; Lailvaux and Irschick, 2006; Rühr et al., 2024; Sakamoto et al., 2019). This was driven methodologically by the development of ad hoc measuring set-ups (Freeman and Lemen, 2008; Herrel et al., 1999; Püffel et al., 2023; Rühr and Blanke, 2022; Sicuro et al., 2021; Weihmann et al., 2015) and conceptually by the ecological and evolutionary relevance of that trait, which was found to relate to diet (Badyaev et al., 2008; Herrel et al., 2005; Santana et al., 2010), ecology (Gomes Rodrigues et al., 2022) and intra-specific conflicts (e.g. intra-sexual competition, inter-sexual niche partitioning; Ginot et al., 2017; Herrel et al., 1999, 2018; Husak et al., 2009). The vast majority of bite force studies focus on maximum voluntary bite force, which is usually measured as the highest force produced by an individual, with limited stimulation from the experimenter, across several trials.

Dissection-derived and 3D-derived biomechanical models of bite forces have been tested against in vivo bite force measurements (Cox et al., 2012; Ginot et al., 2018; Herrel et al., 2008; Püffel et al., 2022 preprint), but they have usually not been tested against each other, making it difficult to assess whether physiological and performance values found in the literature could be compared. In the specific case of insects, to our knowledge no biomechanical model of bite forces based on dissections has been published yet, making it even more uncertain whether these are comparable to the large corpus of studies published on vertebrates, and to the arthropod micro-computed tomography (µCT)-based biomechanics literature. Here, using dissections and 3D reconstructions, we derived various estimates of PCSA for the closer muscles of the mandible of the grasshopper Schistocerca gregaria, which we compared against each other and tested against in vivo measured bite forces.

Specimens

Live adult grasshoppers (n=35, 21 males and 14 females) were obtained from Fressnapf (Krefeld, Germany). The animals were left overnight in their original boxes before any experiment started. Five individuals were directly put in 70% ethanol for practicing dissection of the head, and extraction of the mandible closer muscles. Of the remaining 30 specimens, 15 were dissected and the other 15 used for µCT after the in vivo bite force measurements.

In vivo bite force measurements

The 30 live specimens were measured by one user (S.G.), using a bespoke experimental set-up (Rühr and Blanke, 2022) based on a well-established measuring principle (Herrel et al., 1999). Briefly, the set-up consists of stainless-steel adjustable bite plates, which are mounted on a stand under a stereomicroscope. One of the plates is mobile and transmits compressive forces to a piezo-electric force transducer (model 9215A, Kistler), which converts them into an electrical signal. This signal is in turn amplified by a custom-built amplifier, converted to a digital signal by a 12-bit USB data acquisition device (U3-HV, LabJack Corporation) and passed to the LJStreamUD v1.19 measurement software (LabJack) on a laptop via USB.

To measure bite forces, grasshoppers were held between the thumb and index finger, and their mandibles were opened by anchoring the left mandible to the lower plate (fixed) and pulling upwards lightly to place the right mandible over the upper plate (mobile, and transmitting force to the sensor). Once the bite plates were securely placed between the mandibles, the animals were kept in place without pushing or pulling them. Usually, animals started biting voluntarily. Otherwise, we tried to elicit bites by lightly scratching the head or thorax. If individuals still did not bite, they were left alone for approximately 30 min before a second trial was attempted. Once an individual started biting, it generally kept on biting for long periods. After 3 min, bouts of measurements were discontinued. Overall, 28 specimens bit voluntarily, one specimen bit only on the second trial, and one unmotivated specimen did not bite properly and was excluded from bite force analyses. The space between bite points remained the same for all individuals (ca. 1 mm including the two plates, the thickness of which adds up to 0.55 mm), which showed limited size variation (8.5 mm<head length<10.35 mm). The opening angle was therefore slightly variable between individuals, but we assumed the range of this variation to be much smaller than the optimal range reported for cockroaches, for example, which have a similar size and mandibular mechanics to S. gregaria (58–70 deg; Rühr and Blanke, 2022; Weihmann et al., 2015) (see Fig. S1). This distance was basically the smallest possible with our set-up, ensuring that the state of the muscles when biting would be as close as possible to their state when the animals have their mandibles closed, i.e. the state in which specimens were fixed (see below). Furthermore, we considered a 1 mm thickness to be in the range of objects grasshoppers may bite on in nature (e.g. grasses, leaves, leaf veins). No correction was applied for the possible slight misalignment between the vectors of the bite force and of the force that can be measured, i.e. perpendicular to the bite plate. This misalignment is in any case expected to be relatively small (Püffel et al., 2023), especially as we did not reach opening angles as small as those measured in ants by Püffel and collaborators (2023). The amplifier gain was set to 2 V and all measurements were taken on the same day, at room temperature (20–22°C). Measurements were saved as text files, curated and analyzed in R v4.1.3 (http://www.R-project.org/), using the ad hoc package ForceR, and maximal voluntary bite force was extracted for each individual. Throughout this study, maximal voluntary bite force will be simply referred to as bite force.

Specimen preparation and external measurements

All specimens were euthanized immediately after measuring their bite force, by placing them whole into Falcon tubes filled with a fixative agent (either Bouin solution or 70% ethanol). Out of the 30 specimens, five were kept in 70% ethanol, and were rapidly dissected thereafter for practice, while the rest were fixed in Bouin solution. After the 25 specimens had been fixed in Bouin solution for ∼3 days, 10 were placed in 70% ethanol and rinsed repeatedly, after which they were decapitated and their heads were dissected. The 15 remaining specimens were rinsed, then dehydrated in increasingly concentrated solutions of ethanol, from 70% to 100%, for about 1 h for each step. After this process, the specimens were decapitated and their heads were critical-point dried (Autosamdri 931.GL) before μCT (Bruker SkyScan 1272; voltage 50 kV, current 200 µA, image pixel size 6.0 µm or 7.5 µm) and reconstructed using NRecon. All heads had fully closed mandibles, implying muscle fibers were measured in a contracted state. Before decapitation and before dissection or dehydration for critical point drying, the sex of specimens was determined and five linear measurements (Fig. 1) were collected using a digital caliper (Powercraft, precision 0.01 mm): head height (HH) from the anterior-most point of the frons to the posterior edge of the head (pronotum was partially lifted by the caliper); head length (HL), from the most dorsal point of the vertex to the intersection point of the mandibles; head width (HW), at the widest point between lateral sides (genae) of the head; pronotum width (PW) at the widest point between the lateral sides of the pronotum; and total length (TL), from the most anterior point between the antennas to the most posterior point of the abdomen including cerci.

Fig. 1.

Schematic drawing of a grasshopper, illustrating the linear measurements recorded for all individuals studied here. Original drawing from Wikimedia Commons (CC0).

Fig. 1.

Schematic drawing of a grasshopper, illustrating the linear measurements recorded for all individuals studied here. Original drawing from Wikimedia Commons (CC0).

Dissection, and muscle and fiber measurements

After practicing with the five initial specimens (dissection data from these practice specimens were not included in the analyses of the study), the following dissection sequence was established for the fixed specimens (Fig. 2A). First, the head was carefully cut off the body using fine scissors. Then, under a stereomicroscope, the labium, maxillae, labrum and clypeus were removed, thereby revealing the anterior, ventral and posterior aspects of the mandibles. Additionally, the digestive organs were removed or partially removed by pulling them out with forceps. Generally, during the dissection, the head was held at the corpotentorium or frons with stainless steel fine straight forceps. Alternatively, curved forceps were used to hold the head by pressing lightly on both sides. Cutting generally was achieved using precision Noyes spring scissors. After the external parts were removed, the mandible closer muscles were carefully detached from the cuticle, starting from the posterior side of the head, to detach the posterior and dorsal fibers. In specimens fixated in Bouin solution, fibers could be detached from the cuticle by scratching the inside of the cuticle lightly with the tip of the forceps. Once fibers in a region were detached, scissors were used to cut off and remove that part of the cuticle. Doing this repeatedly allowed removal of the cuticle from the posterior, dorsal and lateral sides of the head capsule. The frons and eyes were removed next, leaving only the closer muscles, tentorium and mandibles (Fig. 2A2). At that point, the tentorium was cut, separating both sides, and removed, revealing the insertion point of the muscle on the mandible. The apodeme (i.e. ‘tendon’ which transfers force from muscle fibers to the mandible) connecting the muscle to the mandible was then cut, as near to the mandibular insertion point as possible to avoid losing muscle fibers, meaning the subsequent muscle mass measurement included the apodeme. A rough estimate of the mass of the apodeme relative to the whole muscle was achieved by computing apodeme volume from one 3D reconstructed specimen (see below). Using an apodeme density of 1.2 mg mm−3, i.e. equivalent to other cuticular structures (Vincent and Wegst, 2004), the apodeme represents about 1% of the total muscle mass. Dissected individual muscles as well as mandibles were then dried using paper towels, and weighed using an OHAUS Explorer analytical balance (readability d=0.1 mg). Muscles were then placed in water in a Petri dish, and individual muscle fibers were dissected from the apodeme. Once most of the fibers were dissected, they were spread out and the Petri dish was photographed (Fig. 2A4) along with millimeter graph paper as the scale, using a Nikon D850 camera with a Nikkor AF-S Micro 60 mm lens, mounted on a Kaiser RA1 copy stand. For each muscle, several hundred fibers were measured with Fiji v1.53f (Schindelin et al., 2012) by drawing straight lines along individual fibers and using the ‘Measure’ tool.

Fig. 2.

Illustration of the sequence of actions applied to dissected specimens and 3D reconstructed specimens of the grasshopper Schistocerca gregaria. (A) Dissection steps and measurements: (1) anterior view of a grasshopper head before dissection; (2) anterior view of the same head after dissection of the head capsule, eyes, nervous system and digestive tract; (3) anterior view of the left closer muscle and mandible after removing the tentorium; (4) part of a photograph of muscle fibers from which length was measured (sides of the squares: 1 cm). (B) 3D image treatment and measurements: (1) lateral view of volume rendering of grasshopper head; (2) lateral view of volume rendering of grasshopper head after removing the head capsule with a custom Fiji macro; (3) result of isosurface extraction and decomposition into non-connected components computed in MorphoDig; (4) mesh of right closer muscle with cuticle; green dots and arrows represent landmarks placed to measure fiber length and muscle force orientation (apodeme length); (5) anterior view of mesh of left and right closer muscles, used to mask the image stack of volume (see 2), thereby producing an image stack with only the muscle of interest (6); this image stack could be imported into Fiji, to produce (7) a stack of 2D convex hulls using a custom Fiji macro. This stack of 2D convex hulls produces a pseudo-3D convex hull (termed ‘2D–3D convex hull’ in the text), which differs from the actual 3D convex hull (8), computed from the mesh of individual muscles in Blender or MorphoDig; (9) oblique view of the mesh (5) in Blender, with a shrink-wrapped plane mesh (orange outline), approximating the insertion area of the muscles on the head capsule.

Fig. 2.

Illustration of the sequence of actions applied to dissected specimens and 3D reconstructed specimens of the grasshopper Schistocerca gregaria. (A) Dissection steps and measurements: (1) anterior view of a grasshopper head before dissection; (2) anterior view of the same head after dissection of the head capsule, eyes, nervous system and digestive tract; (3) anterior view of the left closer muscle and mandible after removing the tentorium; (4) part of a photograph of muscle fibers from which length was measured (sides of the squares: 1 cm). (B) 3D image treatment and measurements: (1) lateral view of volume rendering of grasshopper head; (2) lateral view of volume rendering of grasshopper head after removing the head capsule with a custom Fiji macro; (3) result of isosurface extraction and decomposition into non-connected components computed in MorphoDig; (4) mesh of right closer muscle with cuticle; green dots and arrows represent landmarks placed to measure fiber length and muscle force orientation (apodeme length); (5) anterior view of mesh of left and right closer muscles, used to mask the image stack of volume (see 2), thereby producing an image stack with only the muscle of interest (6); this image stack could be imported into Fiji, to produce (7) a stack of 2D convex hulls using a custom Fiji macro. This stack of 2D convex hulls produces a pseudo-3D convex hull (termed ‘2D–3D convex hull’ in the text), which differs from the actual 3D convex hull (8), computed from the mesh of individual muscles in Blender or MorphoDig; (9) oblique view of the mesh (5) in Blender, with a shrink-wrapped plane mesh (orange outline), approximating the insertion area of the muscles on the head capsule.

3D rendering and 3D data acquisition

Reconstructed μCT image stacks were opened in Fiji and average-binned by a factor of 2 in all dimensions. Image stacks were then opened in MorphoDig v1.6.5 (https://morphomuseum.com/morphodig) to render them as volumes (Fig. 2B). In MorphoDig, isosurfaces were obtained using an individually defined threshold that retained the cuticle, muscle and apodeme, but not softer tissues such as digestive organs. The isosurfaces were cleaned manually and split between left and right muscles using the ‘lasso cut’ tool. On these surfaces, ∼25 muscle fibers per muscle were landmarked, with two landmarks per fiber, one at the intersection between the fiber and apodeme, and the other at the intersection between the fiber and the head capsule. In addition, to define the axis of the force exerted by the muscle, one landmark was placed in the middle of the section of the apodeme near the insertion point of the mandible, and a second one in the trough between the main apodeme and its first posterior ‘wing’ (Fig. 2B4). Landmark coordinates were imported in R, and fiber length was measured using the dist function (package stats) between the respective two landmarks per fiber. Pennation angle was computed with the angle function (Claude, 2008) between the vector of coordinates of the two landmarks placed along the axis of the apodeme and the vector of coordinates of landmarks placed along individual fibers. To measure muscle volume, muscle segmentations were partially automated. First, we developed and applied a custom Fiji macro (available from GitHub, https://github.com/sginot/Grasshopper-PCSA) to all the previously binned image stacks, which removed the external and internal large cuticular structures (mainly the head capsule, mandibles and tentorium; Fig. 2B2). The new image stacks, which were binarized in the previous process using the default settings of Fiji (see Supplementary Information, ‘Sensitivity analyses of muscle volume against binarization algorithms and against decimation (i.e. triangle number)’ and Fig. S2 for comparison of binarization methods), were then imported into MorphoDig, and isosurfaces were again extracted, followed by an in-built MorphoDig function for decomposition into non-connected components, at the same time removing objects smaller than 10,000 triangles (Fig. 2B3). This decomposition generally allowed us to isolate meshes of the closer muscles from other structures, but it was sometimes necessary to manually clean those meshes and separate the left and right muscles. Meshes of individual muscles were then used to compute muscle volume (including apodeme, Fig. 2B6), in Blender (v3.2.0). The 3D convex hull of each muscle mesh was then computed, and the convex hull volume was obtained, using the in-built functions of Blender (Fig. 2B8). Back in MorphoDig, the individual muscle meshes were used as masks on the original volume, to produce binary image stacks of individual muscles. A second custom Fiji macro (available from GitHub, https://github.com/sginot/Grasshopper-PCSA) was then applied to these stacks to convert them into stacks of 2D convex hulls, producing a pseudo-convex hull 3D envelope for the muscle which we term 2D–3D convex hull (Fig. 2B7). This 2D–3D convex hull produces a closer match to the 3D shape of the muscle compared with the normal 3D convex hull, while including the spaces between fibers and fiber bundles, contrary to the direct volume computation of a muscle mesh (which we hereafter refer to as ‘muscle mesh volume’). The hitherto described data acquisition protocols were all applied separately to the left and right mandible closer muscles. Finally, the combined meshes of the left and right closer muscles were imported into Blender, and a new plane mesh was created, extended to the length and width of the muscles, subdivided several times, shrink-wrapped onto the muscle mesh, and adjusted manually in Edit Mode, to create a surface that approximates the area of insertion of the muscle into the cuticle of the head capsule (Fig. 2B9).

3D models were also used to compute the mechanical advantage (MA) of the mandibles. MA determines what proportion of a force applied to a lever is transmitted by that lever. In biological structures, the MA is often smaller than 1, meaning that forces measured in vivo are smaller than the force produced by the muscle contraction. The general definition of MA is MA=IL/OL, where IL is the length of the in-lever, to which the input force is applied, which we define here as the height of the triangle formed by the closer muscle insertion point and the two condyles of the mandible–head hinge joint, which constitute the axis of rotation of the mandible; and OL is the out-lever length, defined here as the height of the triangle between the most distal tip of the mandible and the two condyles. To calculate IL and OL here, four landmarks were placed on each mandible: (i) at the insertion point of the closer muscle apodeme, (ii) at the anterior condyle, (iii) at the posterior condyle, and (iv) at the tip of the most distal ‘tooth’ (or incisivus) of the mandible. IL and OL were then computed as the height of the two triangles formed by the landmarks. The length of the sides of the triangle was computed from landmark coordinates, and the area of the triangle was computed using Heron's formula. Then, the height of the triangles could be obtained from the area and length of the base of the triangle (i.e. the condyle to condyle side). The true value of the in-lever, or effective in-lever (ILeff), is, however, not fixed, but instead depends on the opening angle α of the mandible (Püffel et al., 2023). Here, we define α as the angle between the line joining the incisivus to the axis of rotation (i.e. OL as described previously), and the line joining the right and left axes of rotation. We make the assumption that when the mandibles are fully closed, the IL is aligned with the line between the two axes of rotation, meaning that α is equal to the angle between the IL and OL (α≈55 deg, based on landmark data; Fig. S1). We also assume that in this state, the closer muscle force vector would make a 90 deg angle with the IL, meaning the full force of the muscle is transmitted to the lever. When the mandibles are opened, the angle between the muscle force vector and the IL is not 90 deg, and ILeff, the projection of IL perpendicular to the muscle force vector must be used instead. As we did not record bite forces across a range of angles, we use approximated geometric models (Fig. S1) for α=55 deg and α=85 deg as upper and lower bounds for the effective mechanical advantage (MAeff). The latter value (α=85 deg) was chosen because it was shown to be near the limit of possible bite forces in cockroaches (Weihmann et al., 2015). In this case, we approximate ILeff as ILeff=sin(60 deg)×IL (Fig. S1). We then compute MA=IL/OL for closed mandibles (α=55 deg), and MAeff=ILeff/OL for fully opened mandibles (α=85 deg).

Computation of muscle PCSA

Because the specimens studied here were all fixed with their mandible closed, the computed PCSA represents muscles in a contracted state. This means that the muscle fibers measured here have a larger diameter than when they are in a relaxed state, therefore increasing the PCSA compared with a relaxed muscle. The fiber lengths measured here are also shorter than in a relaxed state, and different from the optimal fiber length (Püffel et al., 2023). The mandible closer muscle is a pennate muscle, for which an effective PCSA (mm2) can be defined as follows (Lieber, 2022; Martin et al., 2020):
(1)
(2)
which corresponds to the PCSA of a non-pennate muscle that would transmit the same amount of force along its axis of action. Here, muscle mass is in g, muscle volume is in mm3 and fiber length is in mm, θ is the mean pennation angle of the muscle fibers and ρ is the muscle density (g mm−3), set to 1.06 g cm−3 a typical value derived for vertebrate muscle (Herrel et al., 2008; Lieber, 2022; Martin et al., 2020), which we expect should be close to that of insects (see Supplementary Information, ‘Measure of mandible closer muscle density in Schistocerca gregaria’). Effective PCSA (PCSAeff) was computed separately for left and right muscles in each specimen. Eqn 1 was used to obtain PCSAeff estimates for dissected specimens. However, pennation angle could not be obtained for these individuals, therefore the per-side average cosθ from 3D reconstructed specimens was used instead of individual values. Eqn 2 was used for 3D reconstructed individuals, also using per-side average cosθ. In these specimens, we had three different proxies for muscle volume (Fig. 2B): (i) muscle mesh volume, (ii) 3D convex hull volume, and (iii) our 2D–3D convex hull volume. These values were all put into Eqn 2, therefore giving three different PCSAeff estimates for all 3D reconstructed specimens. In addition, the insertion area of closer muscles on the head capsule, which was used in previous studies (Goyens et al., 2014), constituted a fourth PCSA proxy for 3D reconstructed individuals. Because we computed the insertion area for left and right muscles together, we divided the obtained area by two to make it comparable to the PCSA estimates for individual muscles computed from 3D volumes or dissections.

PCSA–bite force regression and muscle stress estimation

To compute muscle force from PCSA, one must also know the maximum muscle stress σ. In most of the vertebrate literature, standard values are used (generally 22.5 or 30 N cm−2; Martin et al., 2020). However, work on arthropods has shown that muscle stress values vary widely from species to species and from muscle to muscle (known values range from 8 to 100 N cm−2; Püffel et al., 2023; Weihmann et al., 2015). Because σ is not known for the mandible closer muscles of S. gregaria, bite force could not be estimated from our muscle data. However, the expected relationship between muscle PCSAeff and the force produced (related to in vivo bite force, see below) can allow us to propose estimates of σ. It should, however, be kept in mind that, because fiber length has been measured in a contracted state, it may not match the optimal fiber length, producing non-maximum muscle stress values.

Bite forces measured in vivo (BFin vivo) relate to the muscle force (MF) produced by the right mandible closer muscle (as the right mandible was always the one applied to the mobile bite plate) in the following way: MF=BFin vivo/MA for α=55 deg, and MF=BFin vivo/MAeff for α=85 deg. We therefore calculated the right closer muscle force, and used it in log–log and standard linear regression against PCSA. The slope of the log–log regression of muscle force against PCSAeff is expected to be 1, while the slope of the regression on raw values represents the muscle stress estimate, and its intercept is expected to be 0. In addition, the confidence intervals (CI) associated with slopes and intercepts were used as statistical tests of whether the parameters were different from their expected values and, in the case of muscle stress (i.e. slope of raw values regression), from known stress values from the literature (see section below).

Comparative muscle stress data

To assess whether the various approaches used here gave reasonable estimates of muscle stress, we collected muscle stress data across insect mandible closer muscles from the literature (Goyens et al., 2014; Püffel et al., 2023; Weihmann et al., 2015; Wheater and Evans, 1989) to obtain a range of plausible values. It should be noted that those muscle stress values were obtained across possibly different contractile states, which impacts PCSA measurements. Therefore, these values may not be truly physiologically homologous.

In addition, an independent estimate of muscle stress in S. gregaria was computed based on sarcomere lengths from one of the specimens of this study, using a published regression between sarcomere length and muscle stress (Fig. 3; Taylor, 2000). It should again be noted that the database gathered by Taylor (2000) stems from various earlier sources, which obtained muscle stress and sarcomere length values using many different methods. As such, the taxonomically broad regression produced will necessarily include non-biological signal, which limits the accuracy of our independent muscle stress estimate. From one of our dried and scanned specimens (Sch1), we dissected a few fibers from the left and right mandible closer muscle. We first took pictures of some of these fibers under an Axio Zoom V16 stereomicroscope, reaching a pixel size of 0.35 µm. In areas where fibers were individually isolated, the sarcomeres were directly visible (Fig. 3A). In Fiji, we opened the images and drew straight lines parallel to the main axis of the fibers in the regions showing the sarcomeres clearly (Fig. 3A). We then used the ‘Plot Profile’ tool to obtain the gray value profile along that line, which crossed several sarcomeres perpendicularly. The grayscale profiles were saved and imported into R, where the profiles were plotted and the distances between gray value maxima were manually determined. To confirm the results from this approach, we also CT scanned a few fibers at very high resolution (Fig. 3A), reaching a voxel size of 0.55 µm at the best resolution possible. The reconstructed image stacks were imported in Fiji, cropped to reduce file size, and explored using the ‘Orthogonal Views’ tool. Where sarcomeres were the clearest, we used the same approach as before to extract gray value profiles, and sarcomere length. We therefore obtained several sarcomere lengths for each profile, and generally a few profiles per fiber. We averaged all sarcomere lengths obtained (Fig. 3B), and logged that value. We then used this log average sarcomere length as the x value in the equation of the linear regression line obtained from Taylor's data (Taylor, 2000), to obtain the corresponding y value of log average muscle stress (Fig. 3C). Finally, we applied the exponential function to that log value to obtain the average muscle stress value σ.

Fig. 3.

Sarcomere length and muscle stress. (A) An image of a muscle fiber scanned at high resolution. The alternating light and dark bands are the sarcomeres. The yellow line (length 120µm) along the main axis of the fiber represents the section along which gray values are measured. The extracted gray values profile is shown below; the x-distance between two peaks along the profile is the sarcomere length. (B) Biplot of muscle stress against sarcomere length, based on data from Taylor (2000) (black circles). Red squares show sarcomere lengths obtained from several fibers of one specimen used in this study. The large red square represents the mean for these measured values. (C) The log values were used to predict the log mean muscle stress value, which was then exponentiated to obtain a muscle stress value for our study.

Fig. 3.

Sarcomere length and muscle stress. (A) An image of a muscle fiber scanned at high resolution. The alternating light and dark bands are the sarcomeres. The yellow line (length 120µm) along the main axis of the fiber represents the section along which gray values are measured. The extracted gray values profile is shown below; the x-distance between two peaks along the profile is the sarcomere length. (B) Biplot of muscle stress against sarcomere length, based on data from Taylor (2000) (black circles). Red squares show sarcomere lengths obtained from several fibers of one specimen used in this study. The large red square represents the mean for these measured values. (C) The log values were used to predict the log mean muscle stress value, which was then exponentiated to obtain a muscle stress value for our study.

Sensitivity analysis of muscle mesh volume estimates

To test whether estimates of PCSAeff based on muscle mesh volume would be affected by the image stack treatment, we ran a sensitivity analysis including various binarization algorithms implemented in Fiji. Included in our analysis were the following algorithms: Huang, IJ_IsoData, Intermodes, Max_Entropy, Minimum, Moments, Otsu and Yen. We implemented a custom Fiji macro to semi-automatically produce volume estimates from image stacks of each specimen used here, and which each used all of the binarization algorithms mentioned above. We also tested in a less systematic manner the effect of decreasing the number of triangles in a mesh on volume. To achieve this, we progressively decimated (50%, 80%, 90%) one of our muscle meshes, and computed its volume after each decimation. Results for all analyses are presented in Fig. S2.

Size dependency of shrinkage

Shrinkage due to fixation of specimens is a well-known problem in studies of muscle biomechanics relying on 3D approaches (Hedrick et al., 2018; Vickerton et al., 2013). If shrinkage is consistent across individuals of different sizes, it would cause consistent underestimation of muscle volumes and PCSA, but should not affect the statistical relationship of these muscular traits with forces. However, if shrinkage in inconsistent, or dependent on size, it would generate noisy variation or bias (for example, if larger specimens shrink systematically more than smaller specimens, the variation in muscle volume could be reduced). Therefore, we tested the potential size dependency of shrinkage by computing the amount of shrinkage for individual muscles as the ratio between 2D–3D convex hull volume and muscle mesh volume, which was then compared with size (specifically head width).

Code and analyses

All plots and statistical analyses were done in R v4.1.3 (http://www.R-project.org/). R code and Fiji macros are available from GitHub (https://github.com/sginot/Grasshopper-PCSA).

External morphometric trait variation and in vivo bite force

Maximum voluntary bite force across a sample of 29 grasshoppers ranged from 0.43 to 2.10 N, with a mean (±s.d.) of 1.42±0.40 N (Table 1). Females showed 33% stronger bites than males (Welch two sample t-test, mean=1.65 and 1.23, respectively, t=3.218, d.f.=27, P=0.003). In vivo bite force was significantly correlated with all linear morphometric measurements (linear regression, all R2>0.28, all P<0.01), but HW had the strongest effect on bite force (R2=0.523, P<0.001). Significant sexual dimorphism was found in all linear morphometric measurements (HH, HL, HW, PW, TL), with females being 4% (HH) to 13% (TL) larger than males (Welch two sample t-tests, all P<0.01), depending on the trait. Sexual dimorphism in in vivo bite force was mostly due to size differences between males and females, but the interaction term between HW and sex was also significant (Type II ANOVA; see Supplementary Information, ‘Sexual dimorphism in head size, in vivo bite force and in bite force allometry in Schistocerca gregaria’ and Fig. S3). Bite force scaled with positive allometry to HW, as well as other head traits (see Supplementary Information, ‘Sexual dimorphism in head size, in vivo bite force and in bite force allometry in Schistocerca gregaria’ and Fig. S3), while it scaled isometrically to PW and TL. When sexes were analyzed separately, the smaller sample size meant that relationships between bite force and all measurements were non-significant in females, while in males, bite force scaled isometrically to HH, HL and PW, and had positive allometry against HW and TL. The relatively larger sexual dimorphism of bite force (33% difference) compared with head size or total size measurements (4–13%) is explained by the dimensionality of variables (length versus force, which is proportional to an area) and in some cases positive allometry.

Table 1.

Summary statistics for all raw measurements obtained in this study, combining both sides for muscular measurements

Summary statistics for all raw measurements obtained in this study, combining both sides for muscular measurements
Summary statistics for all raw measurements obtained in this study, combining both sides for muscular measurements

Sarcomere length and muscle stress

The mean (±s.d.) sarcomere length measured across our sample of fibers (Fig. 3A,B), from muscles of different sides, was 7.7±1.4 µm. The log average sarcomere length was therefore 2.0, giving a log average muscle stress value of 6.0 (Fig. 3C). After converting back to a linear scale and converting to N mm−2, we obtained a stress value [95% CI] σ=0.40 N mm−2 [0.17, 0.95 N mm−2]. We also found differences in average sarcomere length between left and right sides (Welch t-test, left 8.9 µm, right 6.4 µm, t=15.988, d.f.=181, P<0.001). It should be noted, however, that this test only reflects a few fibers sampled in a single individual.

Dissected mandible mass, muscle mass, fiber length and PCSAeff

Isolated mandible mass ranged from 0.0035 to 0.0109 g (mean±s.d.=0.0067±0.0021 g). Left and right mandibles were asymmetrical in shape and mass (mean left=0.0074 g, mean right=0.0061 g). Left and right mandible closer muscles were also found to be asymmetrical (visible in Fig. 2A). Therefore, left and right muscles were weighed, and their fiber length measured separately (Fig. 4, Table 2). The right muscles were on average 41% heavier than the left muscles within individuals. Fiber length showed a lesser degree of asymmetry, with right fibers being on average 9% longer than left fibers (Fig. 4A). Finally, despite the fact that right fibers were longer than left fibers (Fig. 4A), which could compensate partly the muscle mass difference (see Eqn 1), left and right muscle PCSAeff values were also asymmetric, with right muscles having on average a 27% larger PCSAeff than left muscles (Fig. 4C). All differences were found to be significant (unpaired t-tests for fiber lengths and paired t-tests for all other measurements; Table 3).

Fig. 4.

Muscle fiber lengths, pennation angles, and physiological cross-sectional areas. (A) Boxplots (showing median, upper and lower quartiles and 1.5× the interquartile range) of fiber lengths for left (orange) and right (purple) muscles, obtained by micro-dissection or by 3D reconstruction and landmarking. (B) Distribution of pennation angles, obtained for 3D reconstructed specimens; vertical dashed lines show the mean values. Colors as in A. (C) Boxplots of physiological cross-sectional area (PCSA) estimates for left and right muscles, using different approaches; note, the specimens dissected are not the same as the 3D reconstructed ones (as illustrated by the dashed line). Colors as in A except the gray boxplot, representing PCSA estimated from muscle insertion, which corresponds to the total muscle insertion (left and right combined), divided by two. Sample sizes are given at the bottom of each panel.

Fig. 4.

Muscle fiber lengths, pennation angles, and physiological cross-sectional areas. (A) Boxplots (showing median, upper and lower quartiles and 1.5× the interquartile range) of fiber lengths for left (orange) and right (purple) muscles, obtained by micro-dissection or by 3D reconstruction and landmarking. (B) Distribution of pennation angles, obtained for 3D reconstructed specimens; vertical dashed lines show the mean values. Colors as in A. (C) Boxplots of physiological cross-sectional area (PCSA) estimates for left and right muscles, using different approaches; note, the specimens dissected are not the same as the 3D reconstructed ones (as illustrated by the dashed line). Colors as in A except the gray boxplot, representing PCSA estimated from muscle insertion, which corresponds to the total muscle insertion (left and right combined), divided by two. Sample sizes are given at the bottom of each panel.

Table 2.

Summary statistics for muscular measurements, detailing left and right measurements

Summary statistics for muscular measurements, detailing left and right measurements
Summary statistics for muscular measurements, detailing left and right measurements
Table 3.

Results of t-tests for left–right asymmetry of the various muscular traits obtained in this study

Results of t-tests for left–right asymmetry of the various muscular traits obtained in this study
Results of t-tests for left–right asymmetry of the various muscular traits obtained in this study

3D reconstructed muscle volume, fiber length, pennation angle and PCSA

The same asymmetry was found for the different muscle volume proxies (Tables 2 and 3). Muscle mesh volume was 33% larger for right than for left muscles; 33% larger for 2D–3D convex hull volume, and 34% larger for 3D convex hull volume. Fiber length, which was measured on the original muscle 3D isosurface, was 4% larger in right compared with left fibers. Pennation angles were on average 18% larger for right muscle fibers than for left fibers, although this may be due mostly to a limited number of fibers with very high pennation angle (Fig. 4B). Finally, PCSAeff estimates also showed asymmetry, which could be expected as the different volume estimates (which are asymmetric as shown earlier) were all combined with the same fiber lengths measurements. PCSAeff estimated from mesh volume was 19% larger in right muscles than in left muscles (Fig. 4C). PCSAeff estimated from 2D–3D convex hull volume was 19% larger in right muscles than in left muscles (Fig. 4C). PCSAeff estimated from 3D convex hull volumes was 20% larger in right muscles than in left muscles (Fig. 4C). All differences were found to be significant (unpaired t-tests for fiber lengths and pennation angles, and paired t-tests for all other measurements; Table 3). An ANOVA was run to test whether the relative difference between left and right muscle PCSAeff was dependent on the method used (i.e. dissection, muscle mesh, 2D–3D convex hull, convex hull), and showed no significant effect of the methods [sum of squares (SS)=0.027, mean square (MS)=0.009, d.f.=3, F=1.161, P=0.333].

Comparison between the various fiber length and PCSA estimates

Dissection and 3D reconstruction are in close agreement in terms of fiber length, with no difference between average left fiber length from dissection and 3D approaches (Fig. 4A, Table 4), and a small but significant difference of about 4% for right fiber lengths (Fig. 4A, Table 4). This was, however, not true for PCSA estimates: taking the dissection-derived estimates as the baseline, muscle mesh 3D volume estimates were around 3 times lower, 2D–3D convex hull volume estimates were around 60% higher, 3D convex hull volume estimates were around 2.5 times higher, and the estimates based on area of insertion, divided by 2 to obtain an average by side, were the highest of all (Fig. 4C, Table 4). All other combinations of mean differences (i.e. the various 3D-based methods against each other) were also tested, and were highly significant (Welch two sample t-tests, all P<<0.001).

Table 4.

Results of t-tests comparing muscle traits values obtained from dissections against the different 3D reconstruction-based methods

Results of t-tests comparing muscle traits values obtained from dissections against the different 3D reconstruction-based methods
Results of t-tests comparing muscle traits values obtained from dissections against the different 3D reconstruction-based methods

Force–PCSA regressions and muscle stress

Right mandible closer muscle force was significantly related to all PCSAeff estimates (Figs 5 and 6), with the exception of PCSAeff values obtained from muscle mesh volume (Figs 5C,D and 6C). When looking at the log–log regressions, none had slopes different from their expected value of 1 (Fig. 6A), and all except the muscle mesh volume log–log regression slope were significantly different from 0 (in other words, there was no significant correlation between log muscle mesh volume and log force). However, while the slope of the dissection-based log–log regression was close to 1 (1.3), slopes for the 2D–3D convex hull, convex hull and insertion area estimates were much higher (2.6–3) (Fig. 6A). Muscle forces computed for opening angle α=85 deg were higher than for α=55 deg, and produced raw regressions with higher slopes (i.e. higher muscle stress). Raw regression intercepts were never significantly different from their expected value of 0 (Fig. 6B); however, the intercept for the dissection-based regression was closer to 0 than all other intercepts. Finally, the slopes of the muscle force–PCSAeff raw regressions, which represent estimates of muscle stress, were all in the range of values known from the literature (from 0.18 to 1.16 N mm−2) (Hedrick et al., 2018; Püffel et al., 2023; Weihmann et al., 2015; Wheater and Evans, 1989), or estimated from sarcomere length (0.41 N mm−2 [0.17, 0.95 N mm−2]) (Fig. 6C). However, the slope for the muscle mesh volume regression was not significantly different from 0 and displayed a 95% CI ([−1.85, 2.29 N mm−2]) which broadly exceeds the expected range for mandible closer muscle.

Fig. 5.

Linear regressions of right mandible closer muscle force (derived from in vivo bite forces) against right mandible closer muscle effective PCSA. PCSA estimates were obtained from (A) dissections, (C) 3D muscle mesh volume, (E) 2D–3D convex hull volume, (G) convex hull volume and (I) half of the total closer muscle insertion area. (B,D,F,H,J) Log–log regressions of the same data. Downward triangles show forces computed for an opening angle of 55 deg (mandibles fully closed), while upward triangles show forces computed for an opening angle of 85 deg (near maximum opening angle). Lines represent the regression lines, with their corresponding formulas (labeled with downward and upward triangles). Parameters of the formulas are accompanied by their 95% confidence interval (CI) in brackets. Shaded areas around the lines represent their 95% CI.

Fig. 5.

Linear regressions of right mandible closer muscle force (derived from in vivo bite forces) against right mandible closer muscle effective PCSA. PCSA estimates were obtained from (A) dissections, (C) 3D muscle mesh volume, (E) 2D–3D convex hull volume, (G) convex hull volume and (I) half of the total closer muscle insertion area. (B,D,F,H,J) Log–log regressions of the same data. Downward triangles show forces computed for an opening angle of 55 deg (mandibles fully closed), while upward triangles show forces computed for an opening angle of 85 deg (near maximum opening angle). Lines represent the regression lines, with their corresponding formulas (labeled with downward and upward triangles). Parameters of the formulas are accompanied by their 95% confidence interval (CI) in brackets. Shaded areas around the lines represent their 95% CI.

Fig. 6.

Comparison of regression parameters with their expected values. Vertical lines represent 95% CI of the parameters. Upward triangles show values for the maximally open mandibles case, and downward triangles are those for the fully closed mandibles. (A) Slopes of the log–log regressions; the expected value is 1, shown by the solid horizontal line. The dashed horizontal line indicates 0. Slopes for the open and closed mandibles are the same, as well as their CI. (B) Intercepts of the regressions; the expected value is 0. (C) Slopes of the regressions, which represent estimates of muscle stress values required to achieve the measured in vivo bite forces, given the different PCSAeff values we computed. These slopes have no theoretically expected value; however, we have represented the range of insect mandible closer muscle stress values known from the literature, shown as the gray shaded area. The solid circle represents the muscle stress value inferred from the sarcomere lengths obtained in one of the specimens of this study, which we passed into a sarcomere length–muscle stress regression from Taylor (2000).

Fig. 6.

Comparison of regression parameters with their expected values. Vertical lines represent 95% CI of the parameters. Upward triangles show values for the maximally open mandibles case, and downward triangles are those for the fully closed mandibles. (A) Slopes of the log–log regressions; the expected value is 1, shown by the solid horizontal line. The dashed horizontal line indicates 0. Slopes for the open and closed mandibles are the same, as well as their CI. (B) Intercepts of the regressions; the expected value is 0. (C) Slopes of the regressions, which represent estimates of muscle stress values required to achieve the measured in vivo bite forces, given the different PCSAeff values we computed. These slopes have no theoretically expected value; however, we have represented the range of insect mandible closer muscle stress values known from the literature, shown as the gray shaded area. The solid circle represents the muscle stress value inferred from the sarcomere lengths obtained in one of the specimens of this study, which we passed into a sarcomere length–muscle stress regression from Taylor (2000).

Size dependency of muscle shrinkage

The ratio of muscle mesh volume to muscle 2D–3D convex hull volume, used as a proxy for shrinkage, ranged from 0.17 to 0.24 (mean±s.d.=0.22±0.02) for the right closer muscle, and from 0.17 to 0.26 (0.22±0.02) for the left closer muscle. In both cases, this ratio was unrelated to head width, used as a proxy for size: linear regressions of the ratio against head width were non-significant for the left (R2=−0.06, F=0.23, d.f.=1, 12, P=0.64) and right (R2=0.01, F=1.10, d.f.=1, 12, P=0.31) muscles.

Our results overall show a decent degree of agreement between 3D biomechanical modeling and dissection-derived data, as well as in vivo data, but also reveal some discrepancies between the various approaches used to compute PCSAeff. Regarding muscular parameters, fiber lengths obtained from micro-dissections and from 3D reconstruction are basically undistinguishable (Fig. 4A). Dissecting fibers did not cause them to shorten, as they were fixed prior to dissection, and the drying step in 3D reconstructed specimens, which could have caused different shrinkage or deformation compared with only chemical fixation (Romeis, 1989), also did not affect fiber length. This result could be expected considering that muscle fibers are firmly attached to rigid structures on both sides via the usual tonofilaments and microtubuli. However, the absence of length differences does not necessarily relate to muscle volume, as the fiber (and therefore muscular) cross-sectional area may have changed under shrinkage. In our study, fiber lengths were manually measured in 3D models, but algorithms are currently being developed to automatize this process (Katzke et al., 2022; Püffel et al., 2023), making it potentially much faster and exhaustive than micro-dissections (Fig. 2A), and also allowing researchers to obtain pennation angles.

3D-based PCSAeff estimates, however, were vastly different from dissection-based estimates (Fig. 4C). PCSAeff estimated from the volume of 3D meshes of muscles were between 2.5 and 3 times smaller than PCSAeff estimated from dissections. This is despite the fact that some fibers might have been lost during dissection. Considering that fiber lengths from 3D and dissection approaches were almost identical, Eqns 1 and 2 may suggest that the overestimated or underestimated PCSAeff can result from (i) wrong computations of the 3D volumes, for example as a result of shrinkage or thresholding of the images, or (ii) an underestimation of the muscle density. The latter can be ruled out by our preparatory investigations (Supplementary Infomation, ‘Measure of mandible closer muscle density in Schistocerca gregaria’), which clearly do not suggest a density 2 or 3 times as large as the typical value used here and elsewhere (1.06 g cm−3). The other possibility is harder to test, and could play a role in the difference between muscle mesh volume-derived and dissection-derived PCSAeff. Considering that muscle mesh volume is much smaller than the 2D–3D convex hull volume, which represents the envelope of the muscle (Figs 2B and 4C), and considering that in the original image stacks, individual fibers are clearly separated from each other by empty spaces (Fig. 2B), it seems at least some amount of shrinkage occurred, which is in agreement with previous studies (Hedrick et al., 2018; Martin et al., 2020; Romeis, 1989; Vickerton et al., 2013).

Three non-exclusive explanations may be put forward regarding the muscle mesh volume low values (Fig. 4, Table 2). (i) The intricate structure of muscles, with individual fibers more or less separated from each other, may be computationally difficult to transfer into a matching mesh, making the volume computation imprecise. Alternatively, (ii) thresholding of the original grayscale images to binary or (iii) chemically induced shrinkage may have led to fiber cross-section reduction, therefore also reducing muscle mesh volume. Such effects would have impacted neither the 2D–3D convex hull- and 3D convex hull-derived estimates, nor the insertion area estimates which all include empty spaces between fibers. Through our a posteriori comparisons of muscle mesh reconstruction methods [see Supplementary Infomation, ‘Sensitivity analyses of muscle volume against binarization algorithms and against decimation (i.e. triangle number)’ and Fig. S2], it can be seen that mesh triangle number should have a limited influence, as surface decimation trials in one muscle suggest small differences even for large reductions in triangle number [i.e. 3.8% reduction in volume for a 90% reduction in triangle number; see Supplementary Infomation, ‘Sensitivity analyses of muscle volume against binarization algorithms and against decimation (i.e. triangle number)’ and Fig. S2]. However, (ii) thresholding does impact the amount of ‘empty’ space between fibers, and therefore the mesh volume estimate [see Supplementary Infomation, ‘Sensitivity analyses of muscle volume against binarization algorithms and against decimation (i.e. triangle number)’ and Fig. S2], but (iii) empty space between fibers is also present in the original grayscale image stack, suggesting chemical shrinkage also has an effect on muscle mesh volume.

While these biases certainly affect the range of muscle mesh volume values and therefore PCSAeff [see Supplementary Infomation, ‘Sensitivity analyses of muscle volume against binarization algorithms and against decimation (i.e. triangle number)’ and Fig. S2], they do not necessarily compromise the correlation between muscle PCSAeff and bite force (Fig. 5B). Our sensitivity analysis [see Supplementary Infomation, ‘Sensitivity analyses of muscle volume against binarization algorithms and against decimation (i.e. triangle number)’ and Fig. S2] shows that using different thresholding algorithms does not produce mesh volumes that are correlated to bite force, suggesting that the absence of correlation is the result of the original muscle shrinkage due to fixation. This absence of correlation might have been explained by the fact that larger specimens tend to undergo more shrinkage than smaller ones, therefore possibly producing similar sized muscles after fixation, while larger individuals had originally produced stronger bite forces. However, no correlation was found between size and shrinkage, computed as the ratio of muscle mesh volume over 2D–3D convex hull volume. This would therefore suggest that the amount of shrinkage represents noisy variation, for example due to slight differences in fixation duration in different individuals, leading to muscle mesh volumes uncorrelated to bite forces. In any case, the absence of correlation between muscle mesh volume and bite force is surprising, as recent data in ants (Püffel et al., 2023) showed good agreement between in vivo and biomechanically estimated bite forces based on muscle mesh volume. The discrepancy between this and our study, which rely on similar methods and models, could be due to the much larger size variation and sample size used by Püffel et al. (2023), or by different levels of shrinkage.

A way to circumvent the problems described in the previous paragraph is to use the convex hull of the muscle mesh. Because the convex hull wraps around all fibers and the apodeme and includes all spaces possibly created by shrinkage, it should not be strongly affected by shrinkage (either chemically or computationally induced by thresholding). Indeed, the results were reversed compared with muscle mesh volume estimates, with 3D convex hull PCSA estimates being twice as large as PCSAeff from dissections (Fig. 4C, Table 4). More importantly, the PCSAeff values produced by this approach do correlate with in vivo measurements (Fig. 5D). Furthermore, the muscle convex hull is very fast and easy to achieve using widely used software (e.g. Blender). One explanation for the discrepancy in the range of values compared with dissection or muscle mesh volume is that the 3D convex hull includes a lot of empty space, not only due to shrinkage but also due to the ‘banana shape’ of the muscle (Fig. 2B).

To avoid too large a PCSAeff overestimation, we developed a ‘2D–3D convex hull’ approach, which consists of a 3D stack of 2D convex hulls based on the muscle image stack (Fig. 2B7). The shape of this 2D–3D convex hull matches the muscle's shape better than the standard convex hull (Fig. 2B). It produces PCSAeff estimates that are still higher than dissection or muscle mesh estimates, but lower than those derived from the standard convex hull (Fig. 4C, Table 4). Finally, the insertion area, used as a proxy for PCSA by previous studies (David et al., 2016; Goyens et al., 2014), produces by far the highest PCSAeff estimates (Fig. 4C, Table 4), suggesting it largely overestimates PCSA.

Despite the large differences in the range of PCSAeff values produced by these different approaches, all of them (except the muscle mesh volume) correlate to the force produced by the muscle (Fig. 5), suggesting they can all serve as proxies for biting performance.

Force–PCSAeff regression agreement with expectations, and muscle stress estimates

All regressions for muscle force against PCSAeff were found to be significant, except for the one derived from muscle mesh volume values (Fig. 5). The reasons for the latter are discussed above. Aside from this case, all log–log regressions have a slope that is not significantly different from the theoretical expectation of 1 (Fig. 6A). In terms of absolute values, the slope of the dissection-derived log–log regression is the closest to 1, while the 2D–3D convex hull, 3D convex hull and insertion area regressions show similar slopes around 2.5–3. In a similar fashion, the intercept of the dissection-derived regression is closer to the expected value of 0 (around −1) than the other regressions, which all have values around −3, although none of them are actually significantly different from 0 (Fig. 6B). Because the dissection-based regression and log–log regression have smaller CI and are closer to the theoretically expected values, it could be argued from our results that the other approaches might produce noisier results, although they all can be used as proxies for muscle forces and bite forces.

Indeed, the slopes of all regressions, which constitute estimates of muscle stress, all fit within the range of values known from the literature for mandible closer muscle stress. Dissection data and 2D–3D convex hull data produce similar muscle stress estimates (∼0.8–0.9 N mm−2; Figs 5A,E and 6) which are in the upper part of the range, but with CI spanning most of it. The 3D convex hull-based estimate is intermediate (∼0.5–0.6 N mm−2; Figs 5G and 6), while the insertion area-based estimate is in the lower part of the range (∼0.3–0.4 N mm−2; Figs 5I and 6). However, the latter stress value is the closest to the one derived from sarcomere length measured in one of our specimens (∼0.4 N mm−2; Fig. 6). In any case, all approaches produce realistic muscle stress estimates, with fairly large CI, which overlap each other (Fig. 6), suggesting no significant differences.

To summarize, we have shown that muscle PCSAeff can be estimated using dissection and various 3D-based approaches. Our results show that most of these methods can serve as proxies for muscle forces and biting performance, with the notable exception of PCSAeff values derived from muscle mesh volume. This latter approach appears to suffer from noisy variation related to fiber cross-section shrinkage due to chemical fixation and/or thresholding. Considering this result, we would argue against the use of this approach in biomechanical and morpho-functional studies, unless one can ensure that shrinkage is limited or absent. Instead, when using 3D data, one can replace muscle mesh volume by muscle 3D convex hull or 2D–3D convex hull, which are less sensitive to shrinkage and do not entail much more work.

Left–right asymmetry in mandibular muscles and function

Although this was not the aim of our study, we found pervasive asymmetry in muscular traits (mass, volume, PCSA, fiber length, sarcomere length). To our knowledge, this asymmetry has not been reported and quantified before in orthopterans, although it has been shown in cockroaches (Weihmann and Wipfler, 2019). These left–right differences also echo the well-described differences in left–right mandible shapes and wear patterns (Chapman, 1964). Taken together, these elements suggest functional differences between the left and right mandibles: the former is longer than the latter, with a smaller closer muscle and shorter fibers, combining to give a smaller PCSA, which may relate to weaker force, but faster and larger movements. It should, however, be noted that sarcomere length appears to be larger in the left closer muscle, compared with the right one, which would instead favor faster fiber contraction in the latter compared with the former (Paul, 2001). Whether this difference in the sarcomeres compensates for the difference in PCSA, possibly leading to equal forces on both sides, cannot be ascertained here. The functional dynamics of left and right mandible movements in orthopterans should be explored further in future studies to understand how this pervasive asymmetry may have been selected for through evolution.

Conclusion

Globally, linking performance and morphology is a major step in studies of biological adaptation (Arnold, 1983). To tackle this, biomechanical models are key, allowing biologists to obtain a deeper understanding of movements and forces produced by organisms and applied to them by external sources. Technological advances now allow morphologists to replace destructive and tedious or impractical dissection work (e.g. for organisms too small or too big) with 3D anatomical reconstructions of ever-increasing detail, which permit the gathering of precise physiological data including fiber length, pennation angle and even sarcomere length (Cox et al., 2012; Gignac et al., 2016; Gröning et al., 2013; Püffel et al., 2021, 2023). PCSA estimations based on dissections and on 3D models had not been compared, which hindered the interpretation of 3D-based models, and their comparison with data from the literature. Our results suggest that comparisons or combinations of biomechanical data from different approaches (e.g. combining 3D and dissection data, or even different 3D approaches in meta-analyses) should be undertaken with care, considering the large differences in the estimates, and the large associated CI. Biomechanical reconstructions relying on other fixation methods, such as iodine staining or hexamethyldisilazane drying may also need to be compared with each other. Future improvements in 3D biomechanical reconstructions will hopefully reduce the discrepancies outlined here, and accelerate data acquisition, thereby opening the way for larger comparative studies.

It is must also be noted that, in addition to methodological discrepancies, PCSA suffers from more fundamental problems with how it is defined in various studies. As pointed out by Püffel et al. (2023), muscle PCSA depends on the contractile state of the muscle, which impacts the length and diameter of muscle fibers. Therefore, even studies which use the same method to estimate muscle properties may not obtain the same PCSA values if the muscle were measured at different contractile states. In practical terms, it may be easiest to measure PCSA always in fully relaxed or fully contracted muscles. Mechanically speaking, however, Püffel et al. (2023) argue that such an approach may not represent truly homologous states for different muscles, because the relationship between optimal fiber length (i.e. the length for which muscle stress is maximal) and fiber length in fully contracted or relaxed muscle, may not be the same in different muscles. Following this line of reasoning, they suggest defining PCSA ‘at an equivalent point of the force–length curve’, for which they propose that the optimal fiber length would be a good option, therefore allowing the community to have a single, comparable value for PCSA, across species and muscles. Although the proposal is sound, it does necessitate additional knowledge about the force–length relationship in individual muscles, which are as of yet not usually included in studies of muscle PCSA and forces.

Agreeing on a common definition for PCSA as well as refining our knowledge about the relationship between different estimation methods are necessary steps to really enable the study of insect muscle biomechanics, which appear more labile than vertebrate muscle (e.g. there is a much wider range of muscle stress values in invertebrates than in vertebrates). Only through these steps can the potential for larger scale comparative studies be unlocked.

We thank Peter T. Rühr and Carina Edel for training with insect bite force measurements and stimulating discussions. Dagmar Wenzel provided technical support for the CT scanning and Christiane Wallnisch supported fixation and preparation of specimens. Andrew Jansen provided advice and tools for dissections.

Author contributions

Conceptualization: S.G., A.B.; Methodology: S.G., A.B.; Software: S.G.; Validation: S.G.; Formal analysis: S.G.; Investigation: S.G., A.B.; Resources: A.B.; Data curation: S.G.; Writing - original draft: S.G., A.B.; Writing - review & editing: S.G., A.B.; Visualization: S.G.; Supervision: A.B.; Project administration: S.G., A.B.; Funding acquisition: A.B.

Funding

This work received support from the European Research Council under grant agreement no. 754290 and the Deutsche Forschungsgemeinschaft under grant agreement number BL 1355/4-1 awarded to A.B.

Data availability

All data needed to evaluate the conclusions in the paper are present in the paper, available from the GitHub repository (https://github.com/sginot/Grasshopper-PCSA) and/or in the supplementary information.

Anderson
,
R. A.
,
McBrayer
,
L. D.
and
Herrel
,
A.
(
2008
).
Bite force in vertebrates : opportunities and caveats for use of a nonpareil whole-animal performance measure
.
Biol. J. Linn. Soc.
93
,
709
-
720
.
Arnold
,
S.
(
1983
).
Morphology, performance and fitness
.
Am. Zool.
23
,
347
-
361
.
Badyaev
,
A. V.
,
Young
,
R. L.
,
Oh
,
K. P.
and
Addison
,
C.
(
2008
).
Evolution on a local scale : developmental, functional, and genetic bases of divergence in bill form and associated changes in song structure between adjacent habitats
.
Evolution
62
,
1951
-
1964
.
Blanke
,
A.
,
Pinheiro
,
M.
,
Watson
,
P.
and
Fagan
,
M.
(
2018
).
A biomechanical analysis of prognathous and orthognathous insect head capsules : evidence for a many-to-one mapping of form to function
.
J. Evol. Biol.
31
,
665
-
674
.
Chapman
,
R. F.
(
1964
).
The structure and wear of the mandibles in some African grasshoppers
.
Proc. Zool. Soc. Lond.
142
,
107
-
122
.
Claude
,
J.
(
2008
).
Morphometrics with R
.
Springer
.
Cox
,
P. G.
,
Rayfield
,
E. J.
,
Fagan
,
M. J.
,
Herrel
,
A.
,
Pataky
,
T. C.
and
Jeffery
,
N.
(
2012
).
Functional evolution of the feeding system in rodents
.
PLoS One
7
,
e36299
.
David
,
S.
,
Funken
,
J.
,
Potthast
,
W.
and
Blanke
,
A.
(
2016
).
Musculoskeletal modelling under an evolutionary perspective : deciphering the role of single muscle regions in closely related insects
.
J. R. Soc. Interface
13
,
20160675
.
Deeming
,
D. C.
,
Harrison
,
S. L.
and
Sutton
,
G. P.
(
2022
).
Inter-relationships among body mass, jaw musculature and bite force in birds
.
J. Zool.
317
,
129
-
137
.
Dickinson
,
E.
,
Davis
,
J.
,
Deutsch
,
A.
,
Patel
,
D.
,
Nijhawan
,
A.
,
Patel
,
M.
,
Blume
,
A.
,
Gannon
,
J.
,
Turcotte
,
C.
,
Walker
,
C.
et al. 
(
2021
).
Evaluating bony predictors of bite force across the order Carnivora
.
J. Morphol.
282
,
1499
-
1513
.
Freeman
,
P. W.
and
Lemen
,
C. A.
(
2008
).
Measuring bite force in small mammals with a piezo-resistive sensor
.
J. Mammal.
89
,
513
-
517
.
Genbrugge
,
A.
,
Herrel
,
A.
,
Boone
,
M.
,
Van Hoorebeke
,
L.
,
Podos
,
J.
,
Dirckx
,
J.
,
Aerts
,
P.
and
Adriaens
,
D.
(
2011
).
The head of the finch : the anatomy of the feeding system in two species of finches (Geospiza fortis and Padda oryzivora): the head of the finch
.
J. Anat.
219
,
676
-
695
.
Gignac
,
P. M.
,
Kley
,
N. J.
,
Clarke
,
J. A.
,
Colbert
,
M. W.
,
Morhardt
,
A. C.
,
Cerio
,
D.
,
Cost
,
I. N.
,
Cox
,
P. G.
,
Daza
,
J. D.
,
Early
,
C. M.
et al. 
(
2016
).
Diffusible iodine-based contrast-enhanced computed tomography (diceCT) : an emerging tool for rapid, high-resolution, 3-D imaging of metazoan soft tissues
.
J. Anat.
228
,
889
-
909
.
Ginot
,
S.
,
Claude
,
J.
,
Perez
,
J.
and
Veyrunes
,
F.
(
2017
).
Sex reversal induces size and performance differences among females of the African pygmy mouse, Mus minutoides
.
J. Exp. Biol.
220
,
1947
-
1951
.
Ginot
,
S.
,
Herrel
,
A.
,
Claude
,
J.
and
Hautier
,
L.
(
2018
).
Skull size and biomechanics are good estimators of in vivo bite force in murid rodents
.
Anat. Rec.
301
,
256
-
266
.
Ginot
,
S.
,
Herrel
,
A.
,
Claude
,
J.
and
Hautier
,
L.
(
2019
).
Morphometric models for estimating bite force in Mus and Rattus : mandible shape and size do better than lever-arm ratios
.
J. Exp. Biol.
222
,
jeb204867
.
Gomes Rodrigues
,
H.
,
Šumbera
,
R.
,
Hautier
,
L.
and
Herrel
,
A.
(
2022
).
Digging up convergence in fossorial rodents: insights into burrowing activity and morpho-functional specializations of the masticatory apparatus
. In
Convergent Evolution
(ed.
V.
Bels
and
A.
Russell
), pp.
37
-
63
.
Springer
.
Goyens
,
J.
,
Dirckx
,
J.
,
Dierick
,
M.
,
Van Hoorebeke
,
L.
and
Aerts
,
P.
(
2014
).
Biomechanical determinants of bite force dimorphism in Cyclommatus metallifer stag beetles
.
J. Exp. Biol.
217
,
1065
-
1071
.
Gröning
,
F.
,
Jones
,
M.
,
Curtis
,
N.
,
Herrel
,
A.
,
O'higgins
,
P.
,
Evans
,
S.
and
Fagan
,
M.
(
2013
).
The importance of accurate muscle modelling for biomechanical analyses : a case study with a lizard skull
.
J. R. Soc. Interface
10
,
20130216
.
Hartstone-Rose
,
A.
,
Dickinson
,
E.
,
Deutsch
,
A. R.
,
Worden
,
N.
and
Hirschkorn
,
G. A.
(
2022
).
Masticatory muscle architectural correlates of dietary diversity in Canidae, Ursidae, and across the order Carnivora
.
Anat. Rec.
305
,
477
-
497
.
Hedrick
,
B. P.
,
Yohe
,
L.
,
Van Der Linden
,
A.
,
Dávalos
,
L. M.
,
Sears
,
K.
,
Sadier
,
A.
,
Rossiter
,
S. J.
,
Davies
,
K. T. J.
and
Dumont
,
E.
(
2018
).
Assessing soft-tissue shrinkage estimates in museum specimens imaged with diffusible iodine-based contrast-enhanced computed tomography (diceCT)
.
Microsc. Microanal.
24
,
284
-
291
.
Heethoff
,
M.
and
Norton
,
R. A.
(
2009
).
A new use for synchrotron X-ray microtomography: three-dimensional biomechanical modeling of chelicerate mouthparts and calculation of theoretical bite forces
.
Invertebr. Biol.
128
,
332
-
339
.
Herrel
,
A.
,
Spithoven
,
L.
,
Van Damme
,
R.
and
De Vree
,
F.
(
1999
).
Sexual dimorphism of head size in Gallotia galloti : testing the niche divergence hypothesis by functional analyses
.
Funct. Ecol.
13
,
289
-
297
.
Herrel
,
A.
,
Podos
,
J.
,
Huber
,
S. K.
and
Hendry
,
A. P.
(
2005
).
Evolution of bite force in Darwin's finches : a key role for head width: bite force in Darwin's finches
.
J. Evol. Biol.
18
,
669
-
675
.
Herrel
,
A.
,
De Smet
,
A.
,
Aguirre
,
L.
and
Aerts
,
P.
(
2008
).
Morphological and mechanical determinants of bite force in bats : do muscles matter?
J. Exp. Biol.
211
,
86
-
91
.
Herrel
,
A.
,
Soons
,
J.
,
Aerts
,
P.
,
Dirckx
,
J.
,
Boone
,
M.
,
Jacobs
,
P.
,
Adriaens
,
D.
and
Podos
,
J.
(
2010
).
Adaptation and function of the bills of Darwin's finches : divergence by feeding type and sex
.
Emu
110
,
39
-
47
.
Herrel
,
A.
,
Petrochic
,
S.
and
Draud
,
M.
(
2018
).
Sexual dimorphism, bite force and diet in the diamondback terrapin
.
J. Zool.
304
,
217
-
224
.
Husak
,
J. F.
,
Lappin
,
A. K.
and
Van Den Bussche
,
R. A.
(
2009
).
The fitness advantage of a high-performance weapon
.
Biol. J. Linn. Soc.
96
,
840
-
845
.
Isip
,
J. E.
,
Jones
,
M. E. H.
and
Cooper
,
N.
(
2022
).
Clade-wide variation in bite-force performance is determined primarily by size, not ecology
.
Proc. R. Soc. B
289
,
20212493
.
Katzke
,
J.
,
Puchenkov
,
P.
,
Stark
,
H.
and
Economo
,
E.
(
2022
).
A roadmap to reconstructing muscle architecture from CT data
.
Integr. Org. Biol.
4
,
obac001
.
Lailvaux
,
S. P.
(
2018
).
Feats of Strength: How Evolution Shapes Animal Athletic Abilities
.
Yale University Press
.
Lailvaux
,
S. P.
and
Irschick
,
D. J.
(
2006
).
A functional perspective on sexual selection : insights and future prospects
.
Anim. Behav.
72
,
263
-
273
.
Lieber
,
R.
(
2022
).
Can we just forget about pennation angle?
J. Biomech.
132
,
110954
.
Martin
,
M. L.
,
Travouillon
,
K. J.
,
Fleming
,
P. A.
and
Warburton
,
N. M.
(
2020
).
Review of the methods used for calculating physiological cross-sectional area (PCSA) for ecological questions
.
J. Morphol.
281
,
778
-
789
.
Mensa
,
F. S.
,
Muzzi
,
M.
,
Spani
,
F.
,
Tromba
,
G.
,
Dullin
,
C.
and
Di Giulio
,
A.
(
2022
).
When the utility of micro-computed tomography collides with insect sample preparation : an entomologist user guide to solve post-processing issues and achieve optimal 3D models
.
Appl. Sci.
12
,
769
.
Paul
,
J.
(
2001
).
Mandible movements in ants
.
Comp. Biochem. Physiol. Part A: Mol. Integr. Physiol.
131
,
7
-
20
.
Powell
,
P.
,
Roy
,
R.
,
Kanim
,
P.
,
Bello
,
M.
and
Edgerton
,
R.
(
1984
).
Predictability of skeletal muscle tension from architectural determinations in guinea pig hindlimbs
.
J. Appl. Physiol.
57
,
1715
-
1721
.
Püffel
,
F.
,
Pouget
,
A.
,
Liu
,
X.
,
Zuber
,
M.
,
van de Kamp
,
T.
,
Roces
,
F.
and
Labonte
,
D.
(
2021
).
Morphological determinants of bite force capacity in insects: a biomechanical analysis of polymorphic leaf-cutter ants
.
J. R. Soc. Interface
18
,
20210424
.
Püffel
,
F.
,
Johnston
,
R.
and
Labonte
,
D.
(
2023
).
A biomechanical model for the relation between bite force and mandibular opening angle in arthropods
.
R. Soc. Open Sci.
10
,
221066
.
Püffel
,
F.
,
Roces
,
F.
and
Labonte
,
D.
(
2023
).
Strong positive allometry of bite force in leaf-cutter ants increases the range of cuttable plant tissues
.
J. Exp. Biol.
226
,
jeb245140
.
Romeis
,
B.
(
1989
).
Mikroskopische Technik
.
Urban & Schwarzenberg
.
Rühr
,
P.
and
Blanke
,
A.
(
2022
).
ForceX and ForceR: a mobile setup and R package to measure and analyse a wide range of animal closing forces
.
Methods Ecol. Evol.
13
,
1938
-
1948
.
Rühr
,
P. T.
,
Edel
,
C.
,
Frenzel
,
M.
and
Blanke
,
A.
(
2024
).
A bite force database of 654 insect species. Scientific Data (in press).
Sakamoto
,
M.
,
Ruta
,
M.
and
Venditti
,
C.
(
2019
).
Extreme and rapid bursts of functional adaptations shape bite force in amniotes
.
Proc. R. Soc. B
286
,
20181932
.
Santana
,
S. E.
,
Dumont
,
E. R.
and
Davis
,
J. L.
(
2010
).
Mechanics of bite force production and its relationship to diet in bats : bite force and diet in bats
.
Funct. Ecol.
24
,
776
-
784
.
Schindelin
,
J.
,
Arganda-Carreras
,
I.
,
Frise
,
E.
,
Kaynig
,
V.
,
Longair
,
M.
,
Pietzsch
,
T.
,
Preibisch
,
S.
,
Rueden
,
C.
,
Saalfeld
,
S.
,
Schmid
,
B.
et al. 
(
2012
).
Fiji : an open-source platform for biological-image analysis
.
Nat. Methods
9
,
676
-
682
.
Sicuro
,
F. L.
,
Oliveira
,
L. F. B.
,
Hendges
,
C. D.
and
Fonseca
,
C.
(
2021
).
Quantifying bite force in coexisting tayassuids and feral suids : a comparison between morphometric functional proxies and in vivo measurements
.
PeerJ
9
,
e11948
.
Taylor
,
G. M.
(
2000
).
Maximum force production : why are crabs so strong?
Proc. R. Soc. Lond. Ser. B: Biol. Sci.
267
,
1475
-
1480
.
Vickerton
,
P.
,
Jarvis
,
J.
and
Jeffery
,
N.
(
2013
).
Concentration-dependent specimen shrinkage in iodine-enhanced microCT
.
J. Anat.
223
,
185
-
193
.
Vincent
,
J.
and
Wegst
,
U.
(
2004
).
Design and mechanical properties of insect cuticle
.
Arthropod. Struct. Dev.
33
,
187
-
199
.
Weihmann
,
T.
and
Wipfler
,
B.
(
2019
).
The generalized feeding apparatus of cockroaches: a model for biting and chewing insects
. In
Insect Mouthparts : Form, Function, Development and Performance
, 1st edn (ed. Harald W. Krenn), pp.
203
-
262
.
Springer International Publishing
.
Weihmann
,
T.
,
Reinhardt
,
L.
,
Weißing
,
K.
,
Siebert
,
T.
and
Wipfler
,
B.
(
2015
).
Fast and powerful : biomechanics and bite forces of the mandibles in the American cockroach Periplaneta americana
.
PLoS One
10
,
e0141226
.
Wheater
,
C. P.
and
Evans
,
M. E. G.
(
1989
).
The mandibular forces and pressures of some predacious Coleoptera
.
J. Insect Physiol.
35
,
815
-
820
.
Wittorski
,
A.
,
Losos
,
J. B.
and
Herrel
,
A.
(
2016
).
Proximate determinants of bite force in Anolis lizards
.
J. Anat.
228
,
85
-
95
.

Competing interests

The authors declare no competing or financial interests.

Supplementary information