Maeva J Orliac
email: maeva.orliac@univ-montp2.fr

Guillaume Billet

Fallen in a dead ear: intralabyrinthine preservation of stapes in fossil artiodactyls

Keywords: inner ear, bony labyrinth, allometry, middle ear ossicles

The stapes is the last of the middle ear ossicle chain and the smallest bone of the mammalian skeleton. Because it rests on the membrane of the fenestra vestibuli during life, the stapes may often fall within the bony labyrinth cavity when soft structures decay after death. In this work, we highlight the unexpected role that the bony labyrinth plays in the preservation of the stapes. Systematic investigation of the bony labyrinth of 50 petrosal bones of extinct and extant artiodactyls led to the discovery of eight cases of "intralabyrinthine" stapes. Three dimensional reconstructions of these stapes allow documenting stapes morphology of four extinct artiodactyl taxa: Microstonyx erymanthius (Suidae), Elomeryx borbonicus (Hippopotamoidea), ?Helohyus plicodon (Helohyidae), and an undetermined Cainotheriidae; and four extant ones Choeropsis and Hippopotamus (Hippopotamidae), and Tayassu and Phacochoerus (Suoidea). ?Helohyus plicodon from the Middle Eocene documents the oldest stapes known for the order Artiodactyla. Morphological study and metric analyses of our sample of artiodactylan stapes show that stapes are likely to carry relevant phylogenetic characters/signal within artiodactyls, and a potential Euungulata signature.

INTRODUCTION

The middle ear ossicles, the malleus, the incus, and the stapes, are contained within the middle ear space. They play a crucial role in sound transmission to the inner ear of mammals by performing an anatomical impedance match that permits sound waves from the air (and the tympanic membrane) to enter the fluid-filled labyrinth (cochlea). The stapes is the most internal of the middle ear ossicle chain and contacts the membrane of the fenestra vestibuli. These ossicles are the smallest bones of the skeleton of modern mammals; their small size in adult therians is due to a large negative ontogenetic allometry [START_REF] Luo | Developmental patterns in Mesozoic evolution of mammal ears[END_REF]. Their size and fragility in therians explain why few middle ear ossicles are found in the fossil record. However, some stapes -the smallest of ear ossicles [START_REF] Nummela | Scaling of the mammalian middle ear[END_REF] have been described from taxa known from the Mesozoic Period for fossil Mammaliaformes (e.g., few fragmentary stapes from Morganucodon and Haldanodon [START_REF] Kermack | The skull of Morganucodon[END_REF][START_REF] Ruf | Reinvestigation of the basicranium of Haldanodon exspectatus (Mammaliaformes, Docodonta)[END_REF]). Several multituberculate stapes have been documented from Palaeocene deposits (e.g., [START_REF] Meng | The stapes of Lambdopsalis bulla (Multituberculata) and transformational analyses on some stapedial features in Mammaliaformes[END_REF][START_REF] Miao | Discovery of three ear ossicles in a multituberculate mammal[END_REF][START_REF] Lillegraven | Cranio-mandibular anatomy of Haldanodon exspectatus (Docodonta; Mammalia) from the Late Jurassic of Portugal and its implications to the evolution of mammalian characters[END_REF][START_REF] Allin | Evolution of the auditory system in Synapsida ("mammal-like reptiles" and primitive mammals) as seen in the fossil record[END_REF][START_REF] Crompton | Relationships of the Liassic mammals Sinoconodon, Morganucodon oehleri, and Dinnetherium[END_REF], and Eutherian stapes have been found in the Late Cretaceous [START_REF] Archibald | Oldest known eutherian stapes and a marsupial petrosal bone from the Late Cretaceous of North America[END_REF]. Given the interest for the first steps of inner ear bones evolution, stapes have been intensively searched for in basal therian groups. On the other hand, relatively few fossil stapes are documented for the crown clade Placentalia. To our knowledge, these include some primates (e.g., the Eocene genera Notharctus and Djebelemur [START_REF] Gregory | On the structure and relations of Notharctus, an American Eocene primate[END_REF]Benoit et al., 2013); Pliocene Homininae [START_REF] Moggi-Cecchi | A fossil stapes from Sterkfontein, South Africa, and the hearing capabilities of early hominids[END_REF]), the Eocene palaenodont Metacheiromys [START_REF] Patterson | The ear region in xenarthrans (= Edentata, Mammalia). Part II. Pilosa (sloths, anteaters), palaeanodonts, and a miscellany[END_REF], a variety of fossil sloths and cingulates (e.g., [START_REF] Woodward | On some remains of Grypotherium (Neomylodori) listai and associated Mammals from a Cavern near Consuelo Cove, Last Hope Inlet, Patagonia[END_REF][START_REF] Guth | Le stapes de quelques édentés fossiles[END_REF][START_REF] Guth | La région temporale des Édentés[END_REF][START_REF] Patterson | The ear region in xenarthrans (= Edentata, Mammalia). Part II. Pilosa (sloths, anteaters), palaeanodonts, and a miscellany[END_REF][START_REF] Blanco | Fossil evidence of frequency range of hearing independent of body size in South American Pleistocene ground sloths (Mammalia, Xenarthra)[END_REF], basilosauroids (e.g., [START_REF] Lancaster | The middle ear of the Archaeoceti[END_REF], and elephantimorphs [START_REF] Ekdale | Morphological variation in the ear region of Pleistocene Elephantimorpha (Mammalia, Proboscidea) from central Texas[END_REF].

The morphology of the stapes varies significantly among species of placental mammals (e.g., as illustrated by [START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF][START_REF] Nummela | Scaling of the mammalian middle ear[END_REF][START_REF] Solntseva | Adaptive features of the middle ear of mammal in ontogeny[END_REF]. It has even been suspected once to reflect major divisions among Placentalia [START_REF] Novacek | Origin and transformation of the mammalian stapes[END_REF], a view which has however been abandoned since (e.g., [START_REF] Patterson | The ear region in xenarthrans (= Edentata, Mammalia). Part II. Pilosa (sloths, anteaters), palaeanodonts, and a miscellany[END_REF]. Still, the phylogenetic signal carried by this bone might be important at least at other taxonomic levels. Pioneering and foundational work on stapes morphology include that of [START_REF] Doran | XVIII. Morphology of the Mammalian Ossicula auditûs[END_REF], [START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF][START_REF] Nummela | Scaling of the mammalian middle ear[END_REF]. In the semantic works of [START_REF] Doran | XVIII. Morphology of the Mammalian Ossicula auditûs[END_REF] and [START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF], the stapes of a dozen of extant artiodactyl taxa are described. Recent developments of micro-CT scan techniques have provided new tools to investigate the ear region of fossil mammals, giving an unprecedented access to its finest structures. During micro-CT scan investigation of the bony labyrinth of fossil artiodactyls one of the authors (MO) noticed numerous cases of post mortem preservation of the stapes in the bony labyrinth, a hollow cavity that shelters fluids and membranous parts of the inner ear organs. A systematic search for stapes trapped in the inner ear cavity was performed on a sample of 50 petrosals of extant and extinct artiodactyls. Among these 50 petrosal bones investigated, eight preserved this ossicle in the bony labyrinth space, indicating a not so rare occurrence of this case. Because it rests on the membrane of the fenestra vestibuli during life, the stapes may often fall within the bony labyrinth cavity when soft structures such as the tympanic membrane decay after death. In this work, we highlight the unexpected role that the bony labyrinth plays in the preservation of the stapes, the smallest and most fragile bone of the mammalian skeleton. We document the anatomy of fossil stapes of four extinct artiodactyl taxa: one Suidae, Microstonyx erymanthius; one Hippopotamoidea, Elomeryx borbonicus; one Helohyidae, ?Helohyus plicodon; and one undetermined Cainotheriidae. ?Helohyus plicodon from the Middle Eocene documents the oldest stapes known for the order Artiodactyla. "Intralabyrinthine stapes" have also been retrieved from bony labyrinths of four extant artiodactyls that had been chemically prepared: Hippopotamidae (Choeropsis; Hippopotamus), and Suoidea (Tayassu, Phacochoerus) and are included in this study. Measurements of the fossil stapes described are used to explore how the dimensions of several aspects of the stapes in Artiodactyla vary relative to one another and relative to the body mass and vestibule volume. Thanks to the availability of outgroup comparative data for the stapedial footplate, a comparison of the artiodactyl condition with that of other placentals could also be performed. The dimensions of the footplate area of the reconstructed stapes also allowed us to calculate low and high frequency limit estimates for the present artiodactyl sample, which adds to the wealth of information provided by this rarely preserved bone.

MATERIAL AND METHODS

Material

Over 50 petrosals of artiodactyls investigated, four fossil and four extant specimens yielded intralabyrinthine stapes. The stapes of the dichobunoid ?Helohyus plicodon is described from the virtual reconstruction of the right in situ bony labyrinth of specimen AMNH 13079 from the Bridger Formation locality B5, Sweetwater County, Wyoming. The stapes of a small undetermined cainotheriid species is described from an isolated right petrosal (UM PDS 3352) from Pech Desse, Quercy, France (MP 28, Late Oligocene, BiochroM'97). That of the hippopotamoid Elomeryx borbonicus is known from a right isolated petrosal (MHNL H 67) from Saint Henri, Bouches du Rhône (MP 26, early Late Oligocene). The stapes of Microstonyx erymanthius is documented from the interior of the bony labyrinth endocast of the mechanically prepared petrosal MNHN.F.MAR3299 illustrated by Orliac (2012: figs. 2-3) from the Late Miocene locality of Maragheh, Iran. Intralabyrinthine stapes have also been retrieved from bony labyrinths of four extant artiodactyls that had been chemically prepared for zoology collections: two hippopotamids Choeropsis liberiensis (UPPal-M09-5-005a) and Hippopotamus amphibius (AMNH-M 24289), and two suoids Tayassu tajaccu (UM N03) and Phacochoerus sp. (AMNH unknown number).

CT scanning and digital endocast extraction

The specimens were scanned using either the high resolution GE phoenix|vtome|x s240 industrial micro-CT scanner at the American Museum of Natural History (New York), the Skyscan 1076 in vivo CT scanner at the ISE-M (UM), or the Viscom X8050-16 CT scanner at the IPHEP. Specimen numbers, CT scan institutions and resolution of each scan is summarized in Table 1. We extracted the digital endocasts of the bony labyrinths and the stapes using the segmentation tools of AVIZO 6.3 (Visualization Sciences Group) and calculated volumes by surface integration. The segmentation process was performed slice-by-slice manually with the limited range only option of the brush tool of AVIZO 6.3. We segmented the vestibules and semicircular canals in two separate label fields to estimate these volumes separately. The resolution of the CT scan data is, for most specimens, sufficient but not optimal to reconstruct an object as small as the stapes. Besides, taphonomic bias such as recrystallization and partial preservation sometimes led us to use different thresholds to reconstruct as much of the structure as possible. Using different thresholds might impact slightly the thickness of some reconstructed parts [START_REF] Gunz | The mammalian bony labyrinth reconsidered, introducing a comprehensive geometric morphometric approach[END_REF] such as the crura (see Fig. 1) that are very thin. However, the general picture and the measured proportions of the stapes are preserved and reliable, except for one specimen (Elomeryx) whose footplate is possibly uncomplete and has been removed from some analyses (see Results). To minimize reconstruction bias, for all specimens of this study, the surface rendering was performed with the "none" smoothing type option.

Measurements

We made linear measurements from the inner ear model using the 3-D measurement tool of AVIZO 6.3. The "vestibule volume" includes the semicircular canals but not the vestibular aqueduct. Measurements of the fossil stapes described here are provided in Table 2.

Relationships between the dimensions of several aspects of the stapes or of one stapedial dimension relative to body mass and vestibule volume were investigated through log/log linear regression analyses calculated with Past [START_REF] Hammer | PAST: paleontological statistics software package for education and data analysis[END_REF]. When the relation was not found to be significant, we subsequently verified if it may however be otherwise when phylogenetic effects are considered. This was done using the ages of divergence and a pattern of relationships constructed from [START_REF] Gatesy | A phylogenetic blueprint for a modern whale[END_REF] in order to perform Phylogenetic Independent Contrasts PICS [START_REF] Felsenstein | Phylogenies and the comparative method[END_REF] with Mesquite module PDAP version 1.14 [START_REF] Maddison | Mesquite: a modular system for evolutionary analysis[END_REF][START_REF] Midford | PDAP package of Mesquite, version 1.14[END_REF]. The cladogram used for this analysis and the branch lengths used are provided in Supplementary Online Material (.nex file).

For a comparison of the variation of the stapes dimensions relative to body mass with taxa outside Artiodactyla, data on the stapedial footplate dimensions in various placentals have been taken from [START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF] only specimens with non-ambiguous specific attribution have been considered here; all measurements and sources for body masses are indicated in Supplementary Online Material). Body mass estimates for fossil artiodactyl taxa have been calculated from total skull length estimates for the small-sized cainotheriid species using [START_REF] Janis | Correlation of cranial and dental variables with body size in ungulates and macropodoids[END_REF] equation and from astragalus dimensions for Elomeryx borbonicus (based on four specimens from St Henri, using the equation of [START_REF] Martinez | The astragalus of Paleogene artiodactyls: comparative morphology, variability and prediction of body mass[END_REF]. Body mass estimates for Microstonyx erymanthius were taken from the NOW database [START_REF] Fortelius | New and Old Worlds Database of Fossil Mammals (NOW)[END_REF], and estimates for ?Helohyus plicodon was taken from [START_REF] Alroy | Cope's Rule and the dynamics of body mass evolution in North American fossil mammals[END_REF].

Abbreviations for collections and institutions

DESCRIPTIONS

The stapes consists of three parts: the head or capitulum, the shaft or body, divided in two crura (the anterior crus and the posterior crus), and the footplate (Fig. 1). The head of the stapes connects to the long arm of the incus, and the footplate of the stapes is kept in the fenestra vestibuli (oval window) of the petrosal with the help of a ring-shaped ligament. The musculus stapedius is attached to the processus muscularis stapedis (Proc. muscularis stapedis), lying in the area of the posterior wall of the tympanum. The shaft of the stapes presents a central cavity which provides passageway, at least in embryos, to the stapedial artery: the foramen intercrurale.

We assume in our sample, that a stapes trapped in a right bony labyrinth is a right stapes (Table 1). Based on illustrations of several mammalian stapes [START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF]), it appears that the Proc. muscularis stapedis is located on the posterior crus. This criteria was primarily used to orientate the few stapes of our sample for which this process could be identified. In our sample, the stapes is asymmetrical, with a crus slenderer than the other. The foramen intercrurale is also wider on one face than the other. On illustrations sourced for extant mammals [START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF][START_REF] Nummela | Scaling of the mammalian middle ear[END_REF] the slender crus can either be the anterior or the posterior one. For the illustrated ungulate mammals (artiodactyls and perissodactyls), however, the slender crus is always the anterior one. We therefore assume here that it is also always the case in artiodactyls. This criterion helps us to orientate the stapes when the Proc. muscularis stapedis could not be identified.

The stapes can be fragile, light, and hollow in some taxa (e.g., Rodentia, Chiroptera), and massive and compact in others (e.g. Carnivora, Cetacea). The stapes of the fossil artiodactyls documented in this study are lightly built; they show a wide range of morphologies but all present a wide foramen intercrurale between the two crura. In some taxa, this foramen is clearly larger on the lateral side of the bone, this is the case in E. borbonicus, H. amphibius, T. tajaccu, and Phacochoerus sp. This character might be an artefact due to a difference of preservation of the bone on the lateral face. A difference of diameter of the foramen intercrurale between both faces is however also observed in other placental mammals [START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF] it is more quadrangular in E. borbonicus, H. amphibius and the suoid species. This difference in shape is mainly due to differences in the antero-posterior width of the head of the stapes capitulum. Half of the stapes of our sample are asymmetrical. As in the case of Helohyus, the two hippopotamid species, and Tayassu, where one crus is thicker than the other (Fig. 2). The internal surface of the crura is concave in all the specimens investigated. The Proc. muscularis stapedis can be identified on all the stapes except for the two extant hippos. In our sample, the distinction between the capitulum of the stapes and the crura is not visible (Fig. 2). This could primarily be due to the less-than-optimal resolution of the reconstructions.

However, in its descriptions of artiodactyl stapes, [START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF] mentions the lack of demarcation of the capitulum in Sus and in Hippopotamus, and a weak demarcation in Camelus.

In turn, according to the same author, the demarcation between the capitulum and the crura is well marked in the Ruminantia Capreolus and Giraffa. The footplates of our artiodactyl sample are oblong with a ratio comprised between 1.56 and 2.27 for the different taxa. Elomeryx borbonicus is remarkable by the slenderness of its footplate (ratio=2.27), although part of it might be missing. Suoidea show a large head compared to the size of the footplate. This is particularly visible in dorsal and posterior views in Fig. 2. Also, the width of the crura is large relative to the width of the footplate and compared to the other taxa of our sample (visible in lateral view Fig. 2).

METRIC ANALYSES

Analyses of relationships between linear measurements taken on the stapes within our artiodactyl sample indicate that the length of the stapedial footplate varies almost isometrically with the length of the stapes (Fig. 3A, Table 2). The same is also true for the footplate width relative to the footplate length, and for the stapes length relative to the vestibule volume (Fig. 3B-C, Table 2). In contrast, the stapes length is clearly negatively allometrically correlated to the body mass (Fig. 3D, Table 2; NB: considering body mass as a three-dimensional variable, the scaling exponent should be ~0.333 to reach isometry, whereas it is 0.196 in our analysis).

Concerning the footplate dimensions, we have been able to gather a large sample including our data in artiodactyls and the measurements given by [START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF] in a number of other placentals and four additional artiodactyl specimens. It includes data for 12 artiodactyl individuals belonging to 11 different species and for 26 non-artiodactyl individuals (Supplementary Online Material). The footplate length and width are both clearly negatively allometric to body mass in this sample (i.e., slope less than one third; see above) (note that, as should be expected, similar results are obtained when plotting the footplate area against the body mass). Interestingly, artiodactyls appear to plot separately from most other placentals (Fig. 4A-B) for both the footplate length and width. The artiodactyl regression line actually indeed differs clearly from the regression line calculated for the sample of other placentals (non-Artiodactyla placentals). Both footplate length and width fit to a regression line that corresponds to a relatively smaller footplate in comparison with the non-Artiodactyla placentals regression line (Fig. 4A-B). The few placental taxa plotting close to the artiodactyl regression line for both length and width of the footplate include the perissodactyls Tapirus and Equus, the capybara and the aardvark.

Finally, the footplate ratio shows no significant relationship to the stapes length in our artiodactyl sample (R²=0.0619, p=0.55), whether it includes or not Elomeryx whose footplate may be missing some lateral part. When taking phylogenetic effects into account (PICs analysis; see Material and Methods), the analysis provides a similar result (R²=0.14; p=0.36) (Supplementary Online Material). This shows that the variation of the footplate ratio is clearly not dependent on the stapes length in our sample. Instead, there might be some phylogenetic influence on the footplate ratio because the highest values (E. borbonicus excepted) are found in the suoid clade (Tayassu (Microstonyx, Phacochoerus)) (values between 1.80 and 1.90). Furthermore, the values of the footplate ratio in artiodactyls (range= 1.56-2.27) are close to the mean of our placental sample excluding artiodactyls (mean=0.55). Therefore, though artiodactyls may be distinguished from lipotyphlan taxa which show a very high ratio (≥2.50), their footplate ratio is not diagnostic when compared to most other placentals in our sample (Supplementary Online Material; Excel file).

DISCUSSION

Frequency and location of « intralabyrinthine » preservation of stapes

Cases of fossil stapes retrieved from intra-labyrinth space occur in several placental groups, but most remain unpublished until now. While several cases have been observed at least in primates (Benoit et al., 2013;R. Lebrun pers. comm.), notoungulates, litopterns, xenarthrans (G. Billet pers. observ.), and proboscideans (A. Schmitt pers. comm.), intralabyrinthine preservation of stapes has not been reported from studies based on a rather wide sample of fossil perissodactyls including equoids (L. Danilo pers. comm.) and rhinocerotids (M. J.

Orliac unpublished data).

The stapes rests on the membrane of the fenestra vestibuli during life; it is connected to this fenestra by an annular ligament [START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF]. Comparison of the stapedial footplate dimensions to the dimensions of the fenestra vestibuli indicates that in most cases in our sample, the latter is much larger than the former (Table 2), making it easy for the stapes to fall in the inner ear space after the soft tissue decays. Artiodactyls present an ossified bulla, and, as such, preservation of in situ stapes, or stapes trapped in the middle ear space, within the bulla would be expected. However, for artiodactyls of our sample that did preserve the auditory bulla, no stapes was recovered in the middle ear cavity but they were recovered in the bony labyrinth of the inner ear. Besides, stapes were recovered more frequently from isolated petrosal (five cases) than from in situ petrosals (two cases). This observation suggests that once it is trapped in the bony labyrinth, a stapes has little chance to fall out of it, even if the cranial remains are displaced/moved/ after death, most probably because of the small size of the fenestrae compared to the dimensions of the stapes (Fig. 5). In half of the cases, the stapes are preserved in the ampullar area (e.g., ?Heloyus, Fig. 5A-C; Microstonyx, Fig. 5J-L); it is otherwise preserved in the elliptical/spherical recesses (e.g., Elomeryx, Fig. 5D-F; Choeropsis) or in the space of the basal turn of the cochlea (e.g., Cainotheriidae indet, Fig. 5G-I).

Stapes dimensions in artiodactyls

Regression analyses on our small artiodactyl sample indicate close isometry between several dimensions of the stapes such as its length and the length and width of its footplate, which denotes that these dimensions can be directly compared using ratios. Moreover, the stapes length is also closely isometrically correlated with the vestibule volume of the bony labyrinth in our sample, whereas it is clearly negatively allometrically correlated to the body mass. This indicates that the stapes and vestibule may present a similar slower growth pace relative to the body mass in artiodactyls. This is not surprising as strong negative allometry has already been shown for both the middle-ear ossicles and the bony labyrinth of the inner ear relative to other cranial elements and/or body mass (e.g., [START_REF] Nummela | Scaling of the mammalian middle ear[END_REF][START_REF] Luo | A new mammaliaform from the early Jurassic and evolution of mammalian characteristics[END_REF][START_REF] Nummela | Scaling of the marsupial middle ear and its functional significance[END_REF][START_REF] Spoor | The primate semicircular canal system and locomotion[END_REF][START_REF] Billet | Petrosal and inner ear anatomy and allometry amongst specimens referred to Litopterna (Placentalia)[END_REF].

In addition, comparison of stapedial footplate dimensions relative to body mass between artiodactyls and other placentals demonstrate that the former group may possess relatively small stapes (or at least small stapedial footplates). Interestingly, the two perissodactyls included in our sample also show a similar pattern to artiodactyls. The relative small size of the stapedial footplate might therefore be a synapomorphy of Euungulata, the clade that gathers Artiodactyla and Perissodactyla (Asher & Helgen, 2010). Moreover, as the stapedial footplate rests on the membrane of the fenestra vestibuli during life, the dimensions of this fenestra may also be reduced in euungulates in comparison to other placentals. This might also be connected to the relative light weight of the malleus and incus in several artiodactyls and perissodactyls [START_REF] Nummela | Scaling of the mammalian middle ear[END_REF]. These preliminary results, which may have significant phylogenetic and potential functional implications (see below hearing estimates based on the footplate area), thus clearly deserve to be investigated in larger samples of placentals in the future.

Finally, we have also shown, based on our small artiodactyl sample, that footplate shape is probably not influenced by allometry in these ungulates. Therefore, footplate shape may be examined in order to detect phylogenetically significant differences between artiodactyl groups. Though a larger sampling is required to confirm it, an elongated stapedial footplate (high footplate ratio) may for example constitute a synapomorphy of the clade Suoidea.

Stapes morphology of Hippopotamidae and amphibious communication

In fully aquatic and semi-aquatic mammals, like cetaceans, sirenians and phocids, underwater hearing implies a modification of the ossicular chain morphology and mass, in relation to bone conduction of sounds (for review see [START_REF] Nummela | Hearing in aquatic mammals[END_REF]. Most of the morphological modifications described in relation to the adaptation of underwater hearing concern the malleus and incus (larger and denser than in terrestrial mammals). Yet, the relation of the mass of the stapes to that of the malleus plus incus is very close to isometry [START_REF] Nummela | Scaling of the mammalian middle ear[END_REF] and the stapes has to be solid enough to stand the inertial force produced by the swing induced by the malleus and incus. The available sample of stapes illustrated for aquatic and semi aquatic mammals show that the stapes of these taxa is massive and presents wide and thick crura, as well as a very small or closed foramen intercrurale (e.g., Phocaena, Balaenoptera, Manatus, Arctocephalus, Otaria, Phoca; see Doran, 1878:Pl. 59, 626-63;Fleisher, 1973:figs 36, 41, 52;).

Hippos are semi aquatic mammals, spending most of their day in the water but feeding mostly on land. With its head in an "amphibious position" (eyes, nostrils and ears above water, mouth and throat below), H. amphibius can respond to sounds emitted underwater: with the ears out of the water, it is able to hear directionally in air and water simultaneously [START_REF] Barklow | Amphibious communication with sound in hippos, Hippopotamus amphibius[END_REF]. Virtually no information is available for C. liberiensis regarding its ability to hear sound directionally underwater, but the morphology of its petrosal bone, with a pachyostotic tegmen tympani also suggests a specialization of sound perception pathway [START_REF] O'leary | Endocranial Strepsirhini: evidence from three primate petrosals from the Eocene of Chambi, Tunisia[END_REF][START_REF] Orliac | Osteological connexions of the petrosal bone of the extant Hippopotamidae Hippopotamus amphibius and Choeropsis liberiensis[END_REF]. The way sound waves are transmitted from water to the inner ear underwater in hippos is still unclear, but it most probably implies bone conduction [START_REF] Barklow | Amphibious communication with sound in hippos, Hippopotamus amphibius[END_REF]. A few stapes of H. amphibius are illustrated in [START_REF] Doran | XVIII. Morphology of the Mammalian Ossicula auditûs[END_REF][START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF]. [START_REF] Doran | XVIII. Morphology of the Mammalian Ossicula auditûs[END_REF] mentions that the three specimens of H. amphibius he could observe present a large foramen intercrurale. It is also the case of the specimen illustrated by [START_REF] Fleischer | Studien am Skelett des Gehörorgans der Säugetiere, einschließlich des Menschen[END_REF] and for the two specimen of Hippopotamidae illustrated in this work. In H. amphibius, as well as in C. liberiensis, the crura are slender when compared to the aquatic and semi aquatic mammals that perform bone conduction cited above. Despite their ability to directionally hear sound underwater, hippos do not show the common morphological modifications of the stapes found in other amphibious mammals.

Auditory capabilities of extant and extinct species

Correlations between middle ear ossicles size and shape, and auditory capacities have been demonstrated in therian mammals (e.g., [START_REF] Rosowski | What did Morganucodon hear?[END_REF][START_REF] Hemilä | What middle ear parameters tell about impedance matching and high frequency petrosal anatomy of Bothriogenys (Mammalia, Artiodactyla, Anthracotheriidae) and petrosal volume and density comparisons among aquatic and terrestrial artiodactyls and outgroups[END_REF][START_REF] Mason | Functional morphology of the middle ear in Chlorotalpa golden moles (Mammalia, Chrysochloridae): predictions from three models[END_REF]Nummela & Sanchez-Villagra, 2006;[START_REF] Nummela | Sound transmission in archaic and modern whales: anatomical adaptations for underwater hearing[END_REF]. Rosowski & Graybeal (1991:fig. 8) used stapedial footplate area to estimate the audible hearing range of the extinct mammaliaform Morganucodon. This correlation has subsequently been used to propose estimates for early placental mammals [START_REF] Meng | Osseous inner ear structures and hearing in early marsupials and placentals[END_REF]. We use here the equations of [START_REF] Rosowski | What did Morganucodon hear?[END_REF] to calculate low and high frequency limit estimates of our artiodactyl sample based on the footplate area of the reconstructed stapes. Results are provided in Table 3.

The lowest values for low frequency limit estimates are found in extant hippos, H. amphibius (low limit = 0.05 kHz, high limit = 15.77 kHz) and C. liberiensis (low limit = 0.10 kHz, high limit = 20.57 kHz). Hearing limits for these taxa have, to our knowledge, never been determined. The extant Hippopotamus amphibius communicates using a variety of sounds, including surface calls with peak frequency at 0.205 ± 0.188 kHz on the surface [START_REF] Barklow | Amphibious communication with sound in hippos, Hippopotamus amphibius[END_REF], a value consistent with the hearing range estimated here for this taxon based on stapes dimensions. H. amphibius also communicates underwater and produce sounds underwater [START_REF] Barklow | Amphibious communication with sound in hippos, Hippopotamus amphibius[END_REF]. The mean peak frequency measured underwater equals 2.087 ± 1.487 kHz [START_REF] Barklow | Amphibious communication with sound in hippos, Hippopotamus amphibius[END_REF], a value which is also consistent with the present estimates. Interestingly, the values found for the extinct smaller hippopotamoid Elomeryx borbonicus (low limit = 0.24 kHz, high limit = 28.09 kHz) are higher than those calculated for extant hippos and correspond to the hearing range estimated for suoid taxa (Table 3). The estimates calculated here for suoid taxa are close to the hearing range of the extant suid Sus scrofa at 60 dB (low limit = 0.04 kHz, high limit = 40.5 kHz, [START_REF] Heffner | Hearing in domestic pigs (Sus scrofa) and goats (Capra hircus)[END_REF], although low frequency limits seem to be overestimated. The small Paleogene artiodactyls ?H. plicodon and the undetermined cainotheriid taxon are the smallest taxa of our sample (see Supplementary Online Material). Their calculated audible hearing range show high upper and lower limits. Their low limit values are superior to that estimated for Diacodexis ilicis based on cochlear coiling (0.543 kHz, [START_REF] Orliac | The inner ear of Diacodexis, the oldest artiodactyl mammal[END_REF]; but closer values are found when [START_REF] Rosowski | What did Morganucodon hear?[END_REF] equation is used for D. ilicis (Table 3). Finding higher limit estimates for smaller taxa is not surprising as the audible range correlates with interaural distance for taxa using binaural cues [START_REF] Heffner | Evolution of sound localization in mammals[END_REF][START_REF] Nummela | Hearing in aquatic mammals[END_REF] and taxa with small heads and close set ears will be able to hear higher frequencies than taxa with large heads. These results indicate that, based on their stapedial footplate area, ?H. plicodon and the undetermined cainotheriid taxon, were most probably not low frequency hearing specialists, and had hearing limits values congruent with their small size .

Footplate area allows calculating estimates of the audible hearing range of extinct taxa. This proxy would need to be complemented by other methods using measurements of the bony labyrinth such as power functions based on basiliar membrane length and cochlear canal coiling that show good correlations with hearing limits [START_REF] West | The relationship of the spiral turns of the cochlea and the length of the basilar membrane to the range of audible frequencies in ground dwelling mammals[END_REF]. As this work focuses on the morphology of the stapes, we chose to use measurements from the stapes only, and not to include data from the bony labyrinth to calculate low and high frequency limit estimates.

CONCLUSIONS

Fossil stapes are likely to be found in a wide array of bony labyrinth of mammals. In artiodactyls, we report a quite frequent occurrence of stapes fallen in the bony labyrinth after death and tissue decay. Stapes are either preserved in the ampulla, in the elliptical/spherical recesses, or in the basal turn of the cochlea. Artiodactyla stapes show diverse morphologies, and stapes are likely to carry potentially relevant phylogenetic characters.

Metric analyses show that, in our sample, the stapes length is closely isometric with the vestibule volume of the bony labyrinth, whereas it is strongly negatively allometric to body mass. Regression analyses between stapedial footplate dimensions and the body mass in a sample of placentals demonstrate that artiodactyls and perissodactyls may possess relatively small stapes compared to other placental mammals. These preliminary results achieved on a reduced sample show that stapedial morphology clearly deserves to be investigated in larger samples of placentals in the future. Benoit, J., E. M. Essid, W. Marzougui, H. Khayati Ammar, R. Lebrun, R. ). Low and high frequency limit estimates are calculated after the equations of [START_REF] Rosowski | What did Morganucodon hear?[END_REF]. * footplate area for D. ilicis calculated after the dimensions of the fenestra vestibuli.

Figure 1 .

 1 Figure 1. Illustration of stapes morphology following the nomenclature of Fleisher (1973).

  Abbreviations for collections and institutionsAMNH, American Museum of Natural History, New York, USA; IPHEP, Institut de Paléoprimatologie, Paléontologie humaine : Evolution et Paléoenvironnements, Poitiers, France; MHNL, Muséum d'Histoire Naturelle de Lyon, Lyon, France ; MNHN, Muséum National d'Histoire Naturelle, Paris, France; UM, Université de Montpellier, Montpellier, France.

Figure 2 .

 2 Figure 2. 3D stapes model reconstructions of Helohyus plicodon (right) (A); Cainotherium sp. (right) (B); Elomeryx borbonicus (right) (C); Choeropsis liberiensis (right) (D); Hippopotamus amphibius (left mirror view) (E); Tayassu tajacu (right) (F); Microstonyx erymanthius (G); Phacochoerus aegypticus (left mirror view) (H), in from left to right medial, posterior, lateral, ventral, and dorsal views. Abbreviation: pm, Proc. muscularis stapedis. Scale bar = 1 mm.

Figure 3 .

 3 Figure 3. Regression analyses between stapes measurements, cube root of vestibule volume and body mass (see data in Table 2) for a sample of eight artiodactyl species. Abbreviations: Ci, Cainotheriidae indet.; Cl, Choeropsis liberiensis; Eb, Elomeryx borbonicus; Hp, ?Helohyus plicodon; Ha, Hippopotamus amphibius; Me, Microstonyx erymanthius; Ph, Phacochoerus; Tt, Tayassu tajacu.

Figure 4 .

 4 Figure 4. Regression analyses between stapedial footplate dimensions (A, footplate length; B, footplate width) and the body mass in a sample of 11 artiodactyl species (12 individuals) and 26 non-artiodactyl placental species (data provided in Supplementary Online Material). Abbreviations: Ez, Equus zebra; Hh, Hydrochoerus hydrochaeris; Oa, Orycteropus afer; Ti, Tapirus indicus.

Figure 5 .

 5 Figure 5. Bony labyrinth and associated stapes of extinct artiodactyl taxa: ?Helohyus plicodon (A-C); Elomeryx borbonicus (D-F); Cainotheriidae indet (G-I); Microstonyx erymanthius (J-L) in anteromedial (A,D,G,J), posterolateral (B, E, H, K), and ventromedial (C, F, I, L) views. Scale bar = 1mm.

Table 1 .

 1 Specimen related and technical parameters. For institution acronyms, see Materials and Methods.

	suprafamilial	taxon	specimen number	side	CT scan	voxel size
	classification				institution	(μm)
	Dichobunoidea	?Helohyus plicodon	USNM 13079	right	AMNH	61.66
		Cainotheriidae indet	PDS 3352	right	UM	35.84
	Hippopotamoidea Elomeryx borbonicus	MHNL H 67	right	UM	35.84
		Choeropsis liberiensis	UPPal-M09-5-005a right	IPHEP	100
		Hippopotamus amphibius	AMNH-M 24289	left	AMNH	120
	Suoidea	Tayassu tajacu	UM N03	right	UM	35.81
	Suoidea	Microstonyx erymanthius	MNHN MAR3299	right	UM	35.81
	Suoidea	Phacochoerus	AMNH-M	left	AMNH	63.94

  . The outlines in lateral/medial views is triangular in ?H. plicodon, the Cainotheriidae indet., and C. liberiensis, whereas

	taxon	fenestra vestibuli (L; W)	stapes length	footplate (L; W)	footplate ratio	foramen intercrurale (L; W)	estimated bodymass	vestibule volume
	?H. plicodon	1.29; 0.91	0.95	0.96; 0.59	1.63	0.46; 0.51	5000	8.24
	Cainotheriidae.	1.34; 0.82	1.21	1.17; 0.75	1.56	0.77; 0.66	1000	8.92
	E. borbonicus	3.09; 1.62	2.5	2.16; 0.95	2.27	1.63; 1.07	105500	44.69
	Hi. amphibius	4.46; 3.40	3.7	3.68; 2.36	1.56	1.02; 0.78	1850000	398.56
	C. liberiensis	3.17; 1.84	2.69	2.83; 1.58	1.79	1.61; 1.36	215000	155.5
	T. tajacu	1.65; 0.9	1.68	1.65; 0.86	1.86	0.86; 0.67	22000	27.2
	Phacochoerus sp.	2.29; 1.19	2.26	1.93; 1.07	1.80	1.26; 0.93	75000	51.38
	M. erymanthius	2.29; 1.65	2.2	2.06; 1.08	1.90	1.5; 1.14	270000	51.38

Table

2

. Measurements of stapes and bony labyrinth of associated petrosal, and body mass estimates of artiodactyl taxa described in the present study. Sources for body mass estimates are provided in the material and method section. All linear measures in mm, body mass in g, volume in mm 3 . Abbreviations: L refers to length and W refers to width.

Table 3 .

 3 Tabuce, and L. Marivaux. 2013. New insights into the ear region anatomy and cranial blood supply of advanced stem Audible hearing range estimates (expressed in kHz at 60 dB SPL re: 20 uPa) for the artiodactyl sample of the present study based on stapedial footplate area (in mm2

	taxon	footplate area	low frequency limit	high frequency limit
	?H. plicodon	0.44	0.98	47.01
	Cainotheriidae indet	0.69	0.6	39.46
	E. borbonicus	1.61	0.24	28.09
	Hi. amphibius	6.82	0.05	15.77
	C. liberiensis	3.51	0.10	20.57
	T. tajacu	1.11	0.36	32.56
	Phacochoerus sp.	1.62	0.23	28.02
	M. erymanthius	1.75	0.22	27.20
	Diacodexis ilicis*	0.58	0.73	42.24

ACKNOWLEDGMENTS

We first want to acknowledge R. O'Leary (AMNH), J. Meng (AMNH), D. Berthet (Musée des Confluences, Lyon), S. Jiquel (UM) for access to the collections they are in charge of. We are grateful to M. Hill (AMNH), F. Guy (IPHEP), and R. Lebrun (UM) for access to CT scan facilities in the American Museum of Natural History, New York, in the Université de Poitiers, France, and in the Université de Montpellier, France, respectively. We are also grateful to L. Danilo, A. Schmitt (MNHN), and R. Lebrun for providing us with information about occurrences of stapes in the material they study. We also want to thank F. Lihoreau (UM) for body mass calculation of Elomeryx borbonicus. This work has been possible thanks to the high-resolution CT scanner facility of the AMNH funded by the NSF MR1-R2 0959384 to N. Landman, D. Ebel and D. Frost, and to the technical facilities of the Montpellier Rio Imaging platform (Montpellier, France) of the labEx CeMEB. We are grateful to the two anonymous reviewers for their fruitful comments. This is ISEM publication ISEM 2015-248.


Tancrède Alméras
email: tancrede.almeras@umontpellier.fr

Barbara Ghislain

Bruno Clair

Amra Secerovic

Gilles Pilate

Meriem Fournier

Quantifying the motor power of trees

Keywords: Biomechanics, Gravitropism, Reaction wood, Maturation strain, Eccentricity, Efficiency

Trees have a motor system to enable upright growth in the field of gravity. This motor function is taken on by reaction wood, a special kind of wood that typically develops in leaning axes and generates mechanical force during its formation, curving up the stem and counteracting the effect of gravity or other mechanical disturbances. Quantifying the mechanical stress induced in wood during maturation is essential to many areas of research ranging from tree architecture to functional genomics. Here, we present a new method for quantifying wood maturation stress. It consists of tilting a tree, tying it to a stake, letting it grow in tilted position, and recording the change in stem curvature that occurs when the stem is released from the stake. A mechanical model is developed to make explicit the link between the change in curvature, maturation strain and morphological traits of the stem section. A parametric study is conducted to analyse how different parameters influence the change in curvature. This method is applied to the estimation of maturation strain in two different datasets. Results show that the method is able to detect genotypic variations in motor power expression. As predicted by the model, we observe that the change in stem curvature is correlated to stem diameter and diameter growth. In contrast, wood maturation strain is independent from these dimensional effects, and is suitable as an intrinsic parameter characterising the magnitude of the plant's gravitropic reaction.

Introduction

Trees are tall slender vertical structures. Their stability is challenged by different external forces, such as wind and gravity. Their mechanical design is adapted to withstand these constraints. This is achieved for example by a strong anchorage and a stiff trunk, made of a stiff and light material, namely wood. These features are part of the tree "skeletal system" [START_REF] Moulia | Posture control and skeletal mechanical acclimation in terrestrial plants: implications for mechanical modeling of plant architecture[END_REF], and are a necessary condition for its stability in the terrestrial environment. This is not, however, a sufficient condition. Growing straight and vertical also involves the action of a "motor system" [START_REF] Moulia | Posture control and skeletal mechanical acclimation in terrestrial plants: implications for mechanical modeling of plant architecture[END_REF]. This is related to the way of growing of the tree. The diameter growth of the stem is achieved by the addition of new wood layers at its external surface. During growth, the load applied on the tree by gravity (i.e. the self-weight) increases at the same time the structure itself increases in size. Because a tree is never perfectly symmetric, the increase in load induces a downward bending movement of the stem. Later growth increments will be added on a bent structure, acting with a larger lever arm, thus inducing further increase in bending moment and change in stem curvature. This situation would always lead to a weeping structure if the effect of gravity was not actively counteracted by the action of a motor system [START_REF] Fournier | Tree biomechanics and growth strategies in the context of forest functional ecology[END_REF]Alméras and Fournier 2009).

Trees are able to induce forces in their wood, in a way similar to the muscles of animals, although the underlying mechanism differs (Alméras and Clair 2016). Normal wood induces forces of low magnitude, while large magnitude is achieved by the production of a special kind of wood, called reaction wood. This wood tends to contract (tension wood) or expend (compression wood) after it is formed. Because this dimensional change is impeded, mechanical stress is induced in wood (Archer 1986). As reaction wood is produced only either on the upper side (tension wood) or on the lower side (compression wood) of a tilted stem, the formation of a new wood layer induces asymmetric distribution of stress. This reaction is generally accompanied with eccentric growth (faster radial growth on the side with reaction wood) and changes in wood elastic properties (Alméras et al. 2005;[START_REF] Clair | Physical and Mechanical Properties of Reaction Wood[END_REF], increasing the asymmetry of forces and thus the bending moment. This results in a bending moment, able to bend up the tree or just compensate for the effect of increasing weight (Alméras and Fournier 2009;[START_REF] Huang | Biomechanical modeling of gravitropic response of branches: roles of asymmetric periphery growth strain versus self-weight bending effect[END_REF].

The magnitude of the mechanical stress induced in wood is quantified through the maturation strain. This parameter can be defined as the strain that would have been induced during wood formation if it were not impeded by surrounding material. Quantifying this parameter is of major importance in different kinds of studies. In forest and wood sciences, the mechanical stresses induced in wood have important economic consequences. They are cumulated over time within the stem (Archer 1986;[START_REF] Kubler | Growth stresses in trees and related wood properties[END_REF], and cause cracks and deformations when felling the tree or sawing the logs. In ecology, this parameter is important to quantify the diversity in this major function of trees [START_REF] Fournier | Integrative biomechanics for tree ecology: beyond wood density and strength[END_REF]Alméras et al. 2009;[START_REF] Clair | Tension wood and opposite wood in 21 tropical rain forest species. 1. Occurrence and efficiency of the G-layer[END_REF] and possible trade-offs with other functions (Alméras et al. 2009). In botany, accounting for the action of tension wood is necessary to understand how the reorientation of axes in involved in the achievement of tree architecture [START_REF] Fisher | Occurence of reaction wood in branches of dicotyledons and its role in tree architecture[END_REF]. In plant physiology, this parameter is relevant to studies of gravitropism [START_REF] Coutand | The gravitropic response of poplar trunks: key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF]Bastien et al. 2013), i.e. the way by which plant perceive and react to gravity, to quantify the response (mechanical stress) to a stimulus (gravity). Tension wood expression has been taken as a model for functional genomics of wood formation [START_REF] Pilate | Tension wood as a model for functional genomics of wood formation[END_REF]). Quantifying tension wood maturation stress is also essential to studies aiming at understanding, at molecular scale, the mechanism by which the plant is able to generate forces (as reviewed in Alméras and Clair 2016).

The usual method for quantifying maturation strain is the released strain method. It consists of releasing the stress at the tree surface and recording subsequent strains. Practically, this is done by setting a LVDT or strain gage [START_REF] Kikata | The effect of lean on growth stress in Pinus densiflora[END_REF] at the stem surface, and cutting two grooves above and below the sensor. The released strain is assumed equal to maturation strain. This assumption is based on the fact that the wood layer at the outermost surface of the stem has been deposited recently. If the trunk is stiff enough and radial growth quick enough, associated increment in external loads has induced negligible stress increment on this layer, so that it is in a nearly native state, i.e. loaded by maturation strain only. Theoretically, this would be true for any tree in active diameter growth, provided the strain is measured at the tree surface and completely released. However, practically, the operation of strain release is made with a sensor of finite length and making grooves of finite depth [START_REF] Yoshida | Techniques for measuring growth stress on the xylem surface using strain and dial gauges[END_REF]. If the grooves are too shallow, not all the stress can be released so that maturation strain is underestimated. If the grooves are too deep, then older layers are also released and the mean state of stress of the probed wood layers is not native. Experiments and calculations show that, because of maturation strains and their accumulation over time, the field of stress in a tree stem has a specific shape [START_REF] Kubler | Growth stresses in trees and related wood properties[END_REF]. In particular, below the tree surface there is a stress gradient, so that the mean stress on a finite depth differs from the surface stress. Moreover, if the changes in tree self-load are quicker than the increase in diameter, as for example due to the development of branches or fruits, the increment of mechanical stress due to external action is no more negligible, so that the outer layer is not in a native state, and the released strain differs from maturation strain.

These metrological issues are particularly important when young stems or branches are studied, as is the case for most studies in a biological context (e.g. Alméras et al. 2006;[START_REF] Clair | Maturation stress generation in poplar tension wood studied by synchrotron radiation microdiffraction[END_REF][START_REF] Coutand | Mechanosensing is involved in the regulation of autostress levels in tension wood[END_REF], Gorschkova et al. 2015[START_REF] Lafarguette | Poplar genes encoding fasciclin-like arabinogalactan proteins are highly expressed in tension wood[END_REF][START_REF] Nishikubo | Xyloglucan endo-transglycosylase (XET) functions in gelatinous layers of tension wood fibers in Poplar-A glimpse into the mechanism of the balancing act of trees[END_REF][START_REF] Roussel | Evidence of the late lignification of the G-layer in Simarouba tension wood, to assist understanding how non-G-layer species produce tensile stress[END_REF]. For small trees, scaling effects due to allometric growth make the stem more sensitive to changes in self-weight, and the finite size of sensors make the experimental method more constraining, so that measurement of maturation strains with the released strain method fails. An alternative method has been proposed to quantify maturation strains during up-righting movement of tilted plant [START_REF] Coutand | The gravitropic response of poplar trunks: key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF]). The method consists of tilting a plant and recording its free up-righting movement. Using a mechanical model, we [START_REF] Fournier | Tree biomechanics and growth strategies in the context of forest functional ecology[END_REF]Alméras and Fournier 2009) made explicit the relation between the variation in curvature (in response to the development of asymmetric maturation strains), dimensions of the section, growth rate, characteristics of the section and reaction wood maturation strain. This model can be used to deduce the maturation strain from the measured change in curvature and the morphological parameters of the stem, as done in [START_REF] Coutand | The gravitropic response of poplar trunks: key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF] and Alméras et al. (2009). This method does not face the same metrological issues as the released strain method, and has the great advantage to be non-destructive as it can be applied during the reaction. It has however some drawbacks. The change in stem curvature on a finite period of time actually depends on the effect of maturation strain and the change in self-weight, and a correction has to be used to cancel the latter (Alméras et al. 2009). Moreover, another problem is that during the up-righting movement of the stem, the reaction is neither constant in space nor constant in time. Indeed, the up-righting movement is accompanied by a straightening movement [START_REF] Coutand | The gravitropic response of poplar trunks: key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF]Bastien et al. 2013) involving production of reaction wood on the opposite side of the distal parts of the stem [START_REF] Coutand | The gravitropic response of poplar trunks: key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF]. Therefore, the model for computing maturation strains has to be applied locally on the stem and for small periods of time, making it heavy to use.

Here, we propose an alternative method, aiming at being more precise and less time-consuming than existing methods. The method consists in tilting a stem, fasten it to a stake, and let it grow in this constrained configuration. At the end of the experiment, the stem is released from the stake and the spring-back movement of the stem is recorded as a change in curvature. Then, a mechanical model can be used to infer the maturation strain from this change in curvature and morphological parameters of the stem (such as it size, growth rate, and eccentricity). The mathematical formulation of this model will be first shown and then used to perform a parametric study analysing the dependence between morphological parameters and change in stem curvature. To demonstrate its potential in comparison with other existing methods, this new method has been applied onto two datasets.

Modelling stem reaction

Summary of this section

In this section, we make explicit the relation between the change in curvature at the level of a stem section, the value of reaction wood maturation strain, and other parameters describing the section, such as initial diameter, diameter increment, eccentricity and variations in elastic modulus. This derivation is shown for three representations of the stem section: a general representation, a model based on sine circumferential variations of properties, and a model based on step variations in properties (as described in next section, Fig. 1). For each representation of the section, two cases will be considered. In the first case, we compute the change in curvature of the stem freely up-righting during growth. In the second case, the stem is restrained during growth by tying it to a stake, and we compute the change in curvature of the stem when releasing it from the stake.

The formulation of the problem is based on beam theory and its application to a growing structure (Archer 1986). For the sine model (similar to [START_REF] Fournier | Tree biomechanics and growth strategies in the context of forest functional ecology[END_REF]Alméras and Fournier 2009), explicit formulae are provided to link maturation strain and curvature, in the free up-righting case (Eq. 27) and in the restrained case (Eq. 31). For the step-variation model, the basic equations enabling numerical computation of the model are provided. The reader not interested in mathematical developments may jump to the "Parametric study" section. Variation in grey level represents sine variations in mechanical properties around the circumference of the section. c Representation with circular shape, step variations in material properties and eccentric growth ring. Grey area represents the sector of reaction wood (here tension wood), with angular extension and specific mechanical properties

Representations of a growing stem section

Three representations will be used to derive the model. The first (Fig. 1a) is suitable for the derivation of a general formulation, and later applied to parametric representations. The second representation (Fig. 1b) is based on that used by Alméras and Fournier (2009), with an eccentric growth ring with sinusoid variations of properties. These sine variations have interesting mathematical properties enabling the development of an explicit analytic formulation, but are not completely realistic to describe circumferential variations in maturation strain. The real situation is not antisymmetric between the side with reaction wood and the opposite side, particularly in the case of tension wood: reaction wood is generally located in a sector, and the circumferential variations in maturation strain are not as smooth as described with the sine model. The last representation (Fig. 1c) is, therefore, considering a sector of reaction wood, and will be used to derive a more accurate representation as described later.

In present study, we illustrate the case of tension wood, but formulations remain correct when compression wood is considered.

General formulation

The formulation of the problem is based on beam theory. As the problem is assumed to have bilateral symmetry, only one component of the curvature (namely around Y, Fig. 1) is considered.

Let us consider a section growing between t and t + t by adding a section increment S . The section is submit- ted to internal loads increment due to impeded maturation strains distributed over the section, and external loads increment inducing a normal resultant force N ext and a bending moment M ext . These loads generate strains and stresses distributed within the section, resulting in an axial contraction 0 and a change in curvature C at the section level.

We consider that wood has elastic behaviour with prescribed strain, so that it follows Hooke's law:

where E is the elastic modulus of the material.

According to Bernoulli hypothesis underlying beam theory, the strain at a given position in the section is:

where x is the distance to the centre of the section and 0 is the strain at the level of the centre of the section.

The condition for static equilibrium is:

( 1) and ( 3) we obtain:

1) = E( -), (2) = 0 -x C, Combining Eqs. (
We consider that the maturation strain is non-zero only on the newly produced wood ring, hence: Combining Eqs. ( 4) and ( 5) we obtain:

We define the axial force and bending moment induced by maturation as:

We define the terms of stiffness of the section as:

(3)

N ext = ∬ S ds M ext = -∬ S x ds.
(4)

N ext = ∬ S E ds -∬ S E ds M ext = -∬ S xE ds + ∬ S xE ds.
(5) = 0 on S = on S.

(6)

N ext = ∬ S E ds -∬ S E ds M ext = -∬ S xE ds + ∬ S xE ds.
(7)

N mat = ∬ S E ds M mat = -∬ S xE ds. ( 8
) K 0 = ∬ S Eds K 1 = -∬ S xEds K 2 = ∬ S x 2 Eds.
Combining Eqs. ( 2), (6-8) and considering an infinitesimal growth increment ( S → dS ) we obtain:

Case of free up-righting

In the case of free up-righting, the external loads are zero:

The elementary variation in curvature can be deduced from ( 9) and ( 10):

The total variation in curvature for a section growing from S 0 to S 1 is obtained by integration:

Case of restrained stem

In the restrained case, the external resultant force and the change in curvature are zero:

The increment in external bending moment can be deduced from ( 9) and ( 13):

The total variation in bending moment for a section growing from S 0 to S 1 can be obtained by integration:

The final change in curvature when releasing the stem from the stake is:

(9)

d 0 dS K 0 + dC dS K 1 = dN ext dS + dN mat dS d 0 dS K 1 + dC dS K 2 = dM ext dS + dM mat dS .
(10)

dN ext dS = 0 dM ext dS = 0.
(11)

dC dS = K 0 dM mat ∕dS -K 1 dN mat ∕dS K 0 K 2 -K 2 1 . (12) ΔC = S 1 ∫ S 0 dC dS dS.
(13)

dN ext dS = 0 dC dS = 0. (14) dM ext dS = K 1 K 0 dN mat dS - dM mat (t) dS . (15) ΔM ext = S 1 ∫ S 0 dM ext dS dS. (16) ΔC = -ΔM ext K 0 (S 1 ) K 0 S 1 K 2 S 1 -K 2 1 S 1
.

Note that this model, although general, neglects the effect of bark, whose stiffness and thickness are generally low.

Simplified analytical model with sine variations of material properties

The first model is based on the representation of a stem section similar to Alméras and Fournier (2009).

The section is assumed circular, with sine variations of material properties: With:

Variations in ring thickness due to eccentric growth are defined by: With:

Combining Eqs. ( 7), (17)(18)(19), the bending moment induced by maturation can be computed as:

(integrals of cosine power functions are provided in Appendix).

Case of free up-righting

Here, we assume that the neutral line is at the geometric centre of the section. Note that this assumption is not consistent with the assumption of heterogeneous elastic modulus, and is potentially a source of error. Making this approximation, K 1 = 0 , Eq. ( 11) reduces to:

(17) 𝛼(𝜃) = ᾱ + Δ𝛼 2 cos (𝜃) = ᾱ 1 + k 𝛼 cos𝜃 , (18) E(𝜃) = Ē + ΔE 2 cos (𝜃) = Ē 1 + k E cos𝜃 . k 𝛼 = 𝛼 2 ᾱ = 𝛼(0) -𝛼(𝜋) 𝛼(0) + 𝛼(𝜋) , k E = E 2 Ē = E(0) -E(𝜋) E(0) + E(𝜋) . (19) 𝛿R(𝜃) = 𝛿 R 1 + k O cos𝜃 . k O = R(0) -R( ) R(0) + R( ) .
(20)

𝛿M mat = -𝛿 R Ē ᾱ 2𝜋 ∫ 0 Rcos𝜃 1 + k O cos𝜃 1 + k E cos𝜃 1 + k 𝛼 cos𝜃 Rd𝜃 = -𝜋R 2 𝛿 R Ē ᾱ k 𝛼 + k E + k O + 3 4 k 𝛼 k E k O . (21) C = M mat K 2 .
For a circular cross section with mean modulus of elasticity Ē , the bending stiffness is given by: Combining Eqs. (20)(21)(22), the curvature increment is:

Considering that D = 2R and 𝛿D = 2𝛿 R , this can be put in the same form as in Alméras and Fournier (2009):

With the form factor f defined as:

From Eq. ( 24) and considering an infinitesimal diameter increment ( D → dD ) it comes that:

The total variation in curvature is obtained after integration:

Case of restrained stem

Assuming the neutral line is at the geometric centre of the section, K 1 (t) = 0 , Eq. ( 14) reduces to:

Considering that D = 2R and 𝛿D = 2𝛿 R and rearranging (20), we obtain: Applying Eq. ( 15), we obtain: Considering (9) and ( 22), the variation in curvature when releasing the stem from the stake is:

(22) K 2 = Ē 𝜋R 4 4 . (23) 𝛿C = -4 ᾱ 𝛿 R R 2 k 𝛼 + k E + k O + 3 4 k 𝛼 k E k O .
(

) C = -4Δ f D D 2 . ( 24
) f = 1 + k E + k O ∕k + 3 4 k E k O . ( 25
) dC dD = - 4Δ f D 2 . ( 26
) ΔC = D 1 ∫ D 0 dC dD dD = 4Δ f 1 D 1 - 1 D 0 . 27
(28) dM ext = -dM mat .

(29)

dM mat dD = - 𝜋 16 D 2 ĒΔ𝛼f . ( 30
) ΔM ext = - D 1 ∫ D 0 dM mat = - 𝜋 16 ĒΔ𝛼f D 3 1 -D 3 0 3 . (31) ΔC = ΔM ext K 2 t 1 = - 4 3 Δ f D 3 1 -D 3 0 D 4 1 .

Model with step distribution of material properties

Reaction wood is here assumed to be located in a sector of the ring, given by angular extension . The remaining of the section is supposed to be made of normal wood. The distribution of material properties E and is assumed uniform over each sector.

The maturation loads are then given as:

S and J are the area and the first moment of area, with subscript RW for reaction wood and NW for normal wood, and indicates their increment. Let I be the second moment of area. For each radius increment R , the increment in area and moments of area relative to the centre of the ring of an eccentric sector limited by angle 1 and 2 are: where 0 , 1 , 2 and 3 are integrals of cosine powers provided in Appendix.

Area and moments of area for the computation of loads and stiffness must be computed relative to a fixed reference. For an eccentric case, the centre of the section is not fixed. The fixed reference is taken at the pith, i.e. the centre of the initial section of radius R 0 . The position of the centre of a ring with radius R relative to the pith is given by: (32)

N mat = ∬ S E ds = RW E RW ∬ S RW ds + NW E NW ∬ S NW ds = RW E RW S RW + NW E NW S NW M mat = -∬ S xE ds = -RW E RW ∬ S RW xds -NW E NW ∬ S NW xds = -RW E RW J RW -NW E NW J. (33) 𝛿S * 𝜃 1 ,𝜃 2 = R𝛿 R 𝜃 2 ∫ 𝜃 1 1 + k O cos𝜃 d𝜃 = R𝛿R 𝜙 0 𝜃 1 , 𝜃 2 + k O 𝜙 1 𝜃 1 , 𝜃 2 𝛿J * 𝜃 1 ,𝜃 2 = R 2 𝛿 R 𝜃 2 ∫ 𝜃 1 cos𝜃 1 + k O cos𝜃 d𝜃 = R 2 𝛿R 𝜙 1 𝜃 1 , 𝜃 2 + k O 𝜙 2 𝜃 1 , 𝜃 2 𝛿I * 𝜃 1 ,𝜃 2 = R 3 𝛿 R 𝜃 2 ∫ 𝜃 1 cos 2 𝜃 1 + k O cos(𝜃) d𝜃 = R 3 𝛿R 𝜙 2 𝜃 1 , 𝜃 2 + k O 𝜙 3 𝜃 1 , 𝜃 2 , ( 34
) x 0 = k O (R -R 0 ).
Increments in moments of area computed at the pith S , J , I can be deduced from those computed at the centre of the section S * , J * , I * : Moment of area for reaction and normal wood sectors can be computed from these equations. The loads (normal force and bending moment) can be deduce using (31).

Stiffness increments for an increment in section are given by: Total stiffness terms K 0 , K 1 , K 2 can be computed by inte- gration. Any radial pattern of variations in modulus of elasticity and eccentricity can be considered with this formulation. In particular, the presence of a pith with finite diameter and negligible stiffness can easily be taken into account.

The variations in curvature in the free up-righting and restrained cases can be deduced from Eqs. (11-12) and (15, 16), respectively.

Model implementation and inversion

Above described models enable the calculation of the variation in curvature ΔC as a function of section's parameters, and in particular the value the maturation strain of reaction wood. For the sine model, the solution can be obtained analytically from Eqs. ( 27) and ( 31). The step-variation model has to be computed numerically. It was implemented using Microsoft Excel and Visual Basic. The estimation of reaction wood maturation strains RW can be obtained by inversion of the model. As the dependence of ΔC in RW is linear, this inversion is trivial.

Parametric study: influence of section parameters on the magnitude of reaction

This parametric study aims on one hand at analysing the influence of the section parameters on the performance of the reaction, namely the variation in curvature, and on the other hand at comparing the result of the analytical sine

S = S * J = J * + x 0 S * I = I * + x 2 0 S * . ( 36
) K 0 = ∬ S Eds = E RW S RW + E NW S NW K 1 = ∬ S xEds = E RW J RW + E NW J NW K 2 = ∬ S x 2 Eds = E RW I RW + E NW I NW .
model and the numerical step model. This analysis will be conducted on the two cases considered, namely free uprighting and restrained stem. The principle of this analysis is that the response to each parameter is computed for fixed value of other parameters. The section parameters are illustrated in Fig. 4. The reference values used for the study (Table 1) are close to the mean value measured in the experimental study described later. For each parameter, the range of variation represents a reasonable range of biologically meaningful values for broad-leaf trees, i.e. trees with tension wood.

Analysis of the influence of sections parameters

Results are shown on Fig. 2. In this section, only the results of the model with step variations in properties (solid lines) will be commented. The comparison with the sine model (dashed lines) will be commented in next section.

Results show that in all cases the variation in curvature is approximately twice lower in the restrained case compared to the free up-righting case. This can be explained considering the difference between the two cases. In the restrained case, the bending moment accumulated during growth is applied on the final structure, with large stiffness. In contrast, in the case of free up-righting, each moment increment is applied to the structure at the time it appears, so that early increment are applied on a structure with lower stiffness, inducing a larger change in curvature.

Figure 2-a shows that the variation in curvature has strong inverse response to the section diameter, as could be predicted from Eqs. ( 27) and (31). Figure 2-b shows that for the free up-righting case the reaction increases with the relative radius increment, due to the fact that variations in curvature are cumulated during growth. By contrast, in the restrained case, the response presents optimum for a relative increment close to 50%. The occurrence of this optimum can be deduced from Eq. ( 31). This has consequences on the practical application of this model, i.e. experimental determination of tension wood maturation strain. Obtaining a maximal reaction by setting a 50% diameter increment improves the accuracy of this measurement as it decreases the relative error on the estimation of curvature.

Figure 2c shows that the response of the curvature is linear in the value of tension wood maturation strain for both cases, as predicted from the equations. This strong response shows that, as previously stated [START_REF] Fournier | Tree biomechanics and growth strategies in the context of forest functional ecology[END_REF], tension wood maturation strain acts as a predominant effect, being the motor of the reaction. The extension of tension wood (Fig. 2d) is also a predominant effect, although it has concave shape in both cases, so that the response is less sensitive for large values of tension wood extension (typically values larger than usually observed).

Other parameters are second-order effects, with less sensitive variations and non-zero values of change in curvature for the minimal values set for the parameter. Effect of eccentricity (Fig. 2e) is linear, and that of the MOE ratio (Fig. 2h) almost linear. In the restrained case, it can be noticed that the effect of MOE ratio is weak, so that small variations in this parameter can be neglected. For the free up-right model, pith dimension may have a large effect if it has large values (Fig. 2f), typically in the case where the stem is very young. This is due to the reduction in stiffness of the stem when pith is large, so that early increments in bending moment act on a compliant structure yielding large changes in curvature. In contrast, pith always has a negligible effect in the restrained case, because its contribution to the stem stiffness after growth is always negligible, due to the dependence between second moment of area and diameter at power 4. Finally, normal wood MOE (for a given TW/NW MOE ratio, Fig. 2g) has no effect on the performance of the reaction, because both the bending moment and the stiffness are proportional to it. Only the MOE ratio (Fig. 2h) plays a role, by increasing the bending moment for a given maturation strain.

Comparison between the sine and step-variation models

The step-variation model will be taken as a reference, because it is more accurate (by taking into account the changes in neutral line when MOE is heterogeneous) and closer to reality (as variations in maturation strain are closer to a step variation than a sine variation).

Figure 2 shows that the sine model always underestimates the curvature compared to the step-variation model. This underestimation of approximately -10% for the restrained case and -20% for the free up-righting case implies an equivalent overestimation when the reverse model is used to estimate the maturation strain of tension wood. The sine model as derived does not reflect the effects of the pith (Fig. 2f) and tension wood extension (Fig. 2d). For other parameters, the trend of the curve is the same as for the step-variation model.

Further parametric analyses (not shown) show that the results of the sine model are exactly equivalent to the stepvariation model when there is no variations in MOE and a tension wood sector of 103° is considered. This critical value actually depends on parameters such as MOE ratio and eccentricity. As stated by Alméras ad Fournier (2009), this equivalence is explained by the fact that the sine model can be considered as the first-order development of a Fourier series describing any variation in properties, for which higher order terms vanish when integrated.

Experimental validation of the method

The method was applied on two different datasets. Results presented in this section aim at validating the method by demonstrating its capacity to discriminate between different plant genotypes and to suppress first-order dimensional effects. The biological interpretation of the results will not be commented further as it is not the focus of this paper.

Plant material

The first study aimed at studying the diversity of the motor function among tropical species. It was conducted on 132 young trees from 16 tropical species. Trees were grown in a greenhouse from seeds or seedlings collected in the forest. The species chosen represent a wide diversity of morphology, their diameter at the end of the experiment ranging from 4 to 16 mm. After being acclimated in a greenhouse, stems were tied to a stake and tilted at an angle of 45° from vertical. Thereafter, trees were grown in tilted position until their external diameter reached approximately 1.5 times the initial diameter to maximize the change in curvature (see section "parametric study"). They reached this diameter after variable time, due to the diversity of growth rates between species. Finally, plants were collected and measured as described in next section.

The second study aimed at unravelling the mechanism of maturation strain generation in tension wood. It was conducted on three different poplar transgenic lines modified for a gene potentially involved in tension wood formation, as well as the wild-type untransformed line, the hybrid Populus tremula x Populus alba clone INRA 717-1B4. Three replicate trees per line were grown from microcuttings in a greenhouse during 4.5 months, until they reached a height of 1.2 m, with little variations between trees. Then they were Fig. 2 Results of the parametric study. Results are given for the two representations (step model and sine model) and for the two cases (free up-righting and restrained reaction). Vertical lines indicates the reference value used when varying other parameters ◂ tied to a stake and tilted at 35° degrees from vertical, grown during 10 weeks in this position and finally collected and measured as described below.

Measurements

The measurement protocol was similar for both studies. Trees were released from the stake, generating an instantaneous spring-back movement, so that the stem was curved, and sometimes passed the vertical (Fig. 3). A segment of the stem located at its middle was then cut and placed in horizontal position on the floor, to supress the effect of self-weight. A picture of the segment was taken to quantify the change in curvature due to stem reaction. Curvature was measured by locating points along the segment using an image analysis software, and fitting them to a polynomial function, from which curvature was derived analytically.

In both studies, a thin section was cut from the middle of the segment to analyse the morphology of the section. All sections showed a sector of tension wood located on the upper part of the section (Fig. 4), with more or less pronounced eccentric growth depending on species. Different parameters were measured to describe the section: diameter of the initial section D 0 (before tilting), final dimension of the section D 1 , diameter of the pith, location of the pith centre (to quantify the eccentricity parameter k O ) and extension of the tension wood sector (Fig. 4). As the maturation strain of normal wood NW is low and has low variability, a value -500 µstrain was assumed for this parameter, close to experimentally measured values [START_REF] Clair | Tension wood and opposite wood in 21 tropical rain forest species. 1. Occurrence and efficiency of the G-layer[END_REF], 2013[START_REF] Fournier | Mesures des déformations résiduelles de croissance à la surface des arbres, en relation avec leur morphologie. Observations sur différentes espèces[END_REF]). Other parameters were set at their measured value. The model is able to discriminate between genotypes

The model with step variation of material properties was used to compute the maturation strain of tension wood ( TW ) from the change in curvature ΔC . Estimated maturation strains ranged between -1300 and -7200 µstrains, with variations between specimens and between species. Oneway analysis of variance evidences significant differences between species for the first study (P < 0.001, Table 2), and between transgenic lines for the second study (P < 0.001, Table 2). This demonstrates the discriminating power of the method and, therefore, its suitability to analyse variations in tension wood maturation strain intensity for various research purposes.

Scaling effects are suppressed by the model

Figure 5a shows the relation between the change in curvature C and computed maturation strain TW . These parameters are correlated (R 2 = 0.48, P < 0.0001) because, as mentioned earlier, the maturation strain has a predominant effect on the change in curvature. However, more than 50% of the variance of the change in curvature is not explained by the maturation strain. Figure 5-b shows that change in curvature is also dependant on the initial radius of the stem R 0 (R 2 = 0.18, P < 0.0001). This is directly related to the size effect on curvature previously mentioned. In contrast, the maturation strain is not correlated to stem diameter (Fig. 5c, R 2 = 0.002, P > 0.05). This demonstrates that the analysis of curvature is not appropriate for estimating the magnitude of the reaction at the material level, because it includes effects of section dimensions that are not directly related to material properties: the smaller the section, the larger the change in curvature, irrespective of the magnitude of maturation strain.

Figure 5-d shows that the change in curvature is practically uncorrelated (R 2 = 0.013, P > 0.05) to the relative radius increment. This is consistent with the weak dependence shown in Fig. 2b. The correlation between maturation strain and relative radius increment (Fig. 5e) is even lower (R 2 < 0.001, P > 0.05), showing that the model captures this weak effect size effect.

Discussion and conclusion

Magnitude of estimated maturation strains

The range of maturation strain values encompass values usually measured with the strain-release method on adult trees [START_REF] Clair | Tension wood and opposite wood in 21 tropical rain forest species. 1. Occurrence and efficiency of the G-layer[END_REF][START_REF] Clair | Patterns of longitudinal and tangential maturation stresses in Eucalyptus nitens plantation trees[END_REF][START_REF] Fournier | Mesures des déformations résiduelles de croissance à la surface des arbres, en relation avec leur morphologie. Observations sur différentes espèces[END_REF]), but values larger than usual (namely larger than ~ 4000 µstrains) are also found. At least two reasons justify that measured values are larger than those measured with the strain-release method on adult trees. First, the strain-release method intrinsically tends to underestimate the magnitude of real maturation strains as argued in introduction. Second, measurements performed in the current study are made on severely tilted and restrained stems, so that the expression of the reaction is strong, and likely larger than in naturally tilted adult trees.

Values measured here are globally two times lower than the values of efficiency estimated with the free up-righting method in Alméras et al. (2009). This is related to the fact that in instance the free up-righting method was used to estimate a global efficiency at the section level, including the effect of form parameters (Eq. 25, Alméras and Fournier 2009;Alméras et al. 2009) rather than the value of maturation strain itself. As form parameter act in synergy with maturation strain, the efficiency at the section level, as described in Alméras et al. (2009), is intrinsically larger than the maturation strains itself.

Estimating efficiency vs. estimating maturation strains

The sine model reflects the effects of variations in MOE and eccentric growth, but does not account for tension wood extension. As a consequence, the value of estimated by the sine model is not exactly equal to the difference in maturation strains between the two sides of the stem. It is the value that would have if the variations were sine variations, and it is equivalent to the step-variation model only if for a particular value of the extension of tension wood. If the extension is larger, as usually the case, estimated by the sine model overestimates the real difference in maturation strain.

The sine model is not able to directly estimate tension wood maturation strain, but is suitable to estimate an efficiency of the reaction as defined in Alméras et al. (2009), i.e. f in Eqs. ( 27) and (31). The efficiency includes both the magnitude of the difference in maturation strain and the effect of form parameters such as eccentricity, distribution of MOE and real extension of tension wood. The same efficiency can be achieved, for example, either by a tension wood with large maturation strain or by a tension wood combining moderate maturation strain with eccentricity and variations in MOE. Compared to the measurement of curvature, it supresses first-order dimensional effects. It quantifies the intensity of the reaction for a given size (diameter) and a given cost (growth). It is suitable e.g. to compare globally the intensity of the reaction between plants, for example to study trade-offs between the motor function and other functions in an ecological context (Alméras et al. 2009), irrespective of the way a given efficiency is achieved.

The step-variation model enables the direct estimation of tension wood maturation strains. This parameter is suitable to compare biomechanical properties of different kinds of woods, for example belonging to different genotypes. Typically, this can be used to estimate the intensity of the tension wood produced in kinematic studies of gravitropism [START_REF] Coutand | The gravitropic response of poplar trunks: key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF]Alméras et al. 2009;Bastien et al. 2013), compare the motor power of different types of tension wood [START_REF] Ruelle | Growth stresses and cellulose structural parameters in tension and normal wood from three tropical rainforest angiosperm species[END_REF] or to study the mechanism generating the mechanical stress during tension wood formation (reviewed in Alméras and Clair 2016).

The restrained stem method vs. the free up-righting method

The restrained stem method has many advantages on the free up-righting method. The free up-righting movement of a stem is the result of two phenomena: the gravitropic up-righting and the autotropic straightening [START_REF] Coutand | The gravitropic response of poplar trunks: key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF]Bastien et al. 2013). The former is expressed in early stages of the uprighting process, and implies the production of tension wood in the upper side of the stem. The latter occurs in a second phase, and implies an inversion of stem curvature necessary to achieve a straight and vertical shape without overshooting the vertical. The stimulus (inclination) and the reaction are, therefore, not constant in time (during growth), neither in space (along the stem). This implies that the model can only be applied over short periods of time, and only on parts of the stem that are actively reacting. Thus, changes in curvature have to be repeatedly measured, and the model has to be applied on short growth increments. In contrast, during the growth of a restrained stem, the stimulus and reaction are kept constant in space and time, so that this method can be applied over the whole growth period. Less measurements are thus necessary and the method is more accurate.

Moreover, because the method is applied to the full period of growth, the morphological and physical properties of the section (initial diameter without bark, growth increment, eccentricity, tension wood extension, variations in MOE) obtained after stem sectioning can be used to feed the model. Furthermore, in the free up-righting method, the change in self-weight during up-righting has to be accounted for (Alméras et al. 2009), whereas in the case of the restrained stem it can be neglected. Finally, the effect of maturation kinetics [START_REF] Coutand | The gravitropic response of poplar trunks: key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF], i.e. the delay between tension wood formation and induction of maturation strains may have consequences in the case of the free up-righting method, whereas these consequences vanish when a large period of time is considered, as for the restrained stem method. The restrained stem method need less measurements and is more precise than the free-up-righting method to estimate the efficiency in the motor function. Application of this method to various research contexts should enable significant progress on the understanding of the motor function of trees.

Fig. 1

 1 Fig. 1 Representations of a growing stem section. a Section of any shape with bilateral symmetry. The initial section is S and the section increment is S . Variation in grey level represents any variations in mechanical properties around the circumference of the section. Tissues with larger tension on the right side generate a bending moment M whose sign is indicated on the figure. b Representation with circular shape and sine variations in material properties. The initial section has diameter D and the newly grown ring D + D . The growth

Fig. 3 Fig. 4

 34 Fig.3Illustration of the tree reaction. Left: stem tied to a stake that has grown tilted. Right: spring-back movement after releasing the stem from the stake

Fig. 5

 5 Fig. 5 Suppression of dimensional effects by the model: the change in curvature is strongly correlated to initial diameter and slightly correlated with relative growth increment. Computation of tension wood maturation strain suppresses these correlations

Table 1

 1 Range of values used for the parametric study. TW: tension wood, NW: normal wood, MOE: modulus of elasticity

		Unit	Reference value Range
	Initial diameter	mm	5	1-10
	Relative diameter	-	0.5	0-3
	increment
	Pith relative radius	-	0.4	0-0.95
	TW extension	°135	0-180
	Eccentricity	-	0.67	0-0.95
	NW MOE	MPa	2800	1000-10,000
	TW/NW MOE	-	1.2	0.5-2
	Maturation strain TW µstrain -5000	-500 to -10,000

Table 2

 2

	Analyses of variance of the effect of genotype on	Source	DOF	Sum Sq	Mean Sq	F	Pr > F
	tension wood maturation strain	First study (Poplar transgenic lines)
		Model	4	7.60E-06	1.90E-06	5.330	0.001
		Error	46	1.64E-05	3.56E-07
		Total	50	2.40E-05
		Second study (Tropical diversity)
		Model	15	5.51E-04	3.67E-05	10.951	< 0.0001
		Error	116	3.89E-04	3.36E-06
		Total	131	9.40E-04

Acknowledgements This work was done in the framework of the project 'StressInTrees' funded by the French National Research Agency (ANR-12-BS09-0004).

Compliance with ethical standards

Author contributions statement TA developed the model and wrote the paper. MF had the original idea and developed the model. BC and BG made the measurement for the study on tension wood tropical diversity. GP and AS made the measurements for the study on poplar transgenic lines.

Conflict of interest

The authors declare that they have no conflict of interest.

Appendix

Integrals of cosine powers: