Influence of neck flexibility on the feeding envelope.

<p>A) A flexible neck with limited excursion angles allows harvesting of a sector (yellow) of the theoretically possible entire feeding envelope. B) Free excursions at the basis of an otherwise stiff and inflexible neck give access to only a peripheral part of the entire potential feeding envelope. C) Long-necked Canadian geese can and do flex their necks freely. In relaxed resting as well as in watching positions, the necks are kept upright. Both neck positions keep energy requirements low. While feeding, birds usually reduce the bending moments acting along their necks by assuming a sigmoid neck posture: near the trunk the usual downward convexity, near the head a convexity directed upward. These curvatures of the neck reduce the lever lengths, specifically the distances between the neck base and the segment weights contained in the neck. Abbreviations: b  =  forelimb length, n  =  neck length, d  =  the distance c overed during a given time. All these values are of the same size in A and B.</p

supplementary 2

List of taxa used for forelimb ratios according to supplementary data of Joyce and Gauthier (2004) with added taxa

supplementary file 1

Table of per cent Limb ratios of various extant cryptodire and extinct eucryptodire turtles. Higher aquatic adaptation to the right

View from above (top) on sauropods, which flex their necks laterally.

<p>A – C) The proximal neck segments exposed to torsion are marked by a heavy long axis. The dots are the CoM (centers of mass) of the head plus neck segments distal to the flexed joints. The masses concentrated in the CoM (CoM 1, 3 or 5, respectively) become smaller with a distal shift of the flexure and their lever arms (l₁ in comparison to l₃ or l₅) become shorter. A) The moment of the heavy and long neck is so great, that the inner (right) foot must be placed laterally in order to expand the area of support and to prevent imbalance of the whole animal. B) The same is shown for flexion of the neck to the left. C) The rotating moment is so small that it does not require a lateral placement of a forefoot. D – F) Torsional moments evoked by lateral flexion remain constant along the posterior part of the neck. In all cases shown here, the tails are flexed into the direction opposite to the neck. So the imbalance caused by lateral flexion of the neck can be reduced. The degree to which the tail can be used to counterbalance the neck depends from the ratio CoM 1 * l₁/CoM2 * l₂, or CoM3 * l₃/CoM4 * l₄, respectively. G) Maximal torsional moments that can occur along the neck from segment 2 – segment 16.</p

Simplified model of a sauropod dinosaur: A heavy beam on two pairs of support (limbs).

<p>The bending moments vary dependent from the lengths of the segments (A and B), dependent from the mass distribution (A and C), and dependent from the inclination of the cantilevers at both ends (A and D). The current study is focused on the cantilever segments (dark grey). L is full neck or tail length, l indicates the lever lengths of segment weights.</p

Proximal part of a schematic neck seen from on top illustrates flexibility.

<p>A) Elongation of the segments (cervical vertebrae) makes the radius of curvature longer. Note that the cervical ribs do not contact vertebrae because they deviate ventrally from the axes of the centra. B) The segments (cervical vertebrae) 12 – 18 are deflected by 20° each. This corresponds to the lateral deflection observed by Dzemski (2006) in the ostrich. In addition, (long) cervical ribs are shown on both sides of the vertebrae. C) Shortening of the segments leads to a sharper curvature of the neck.</p

External equilibrium of a sauropod, depending on neck posture.

<p>External equilibrium is determined by the moments of segment weights about the hind feet, which must be equal to the ground reaction force F_v1 or F_v2, respectively, exerted by the forefeet with a lever arm l_f (F_v * l_f). Note that the tail exerts a nose up-rotating torque, because of its negative lever arm (1₇) Low neck position gives the weight forces of head and neck (F₁, F₂) long lever arms (1₁, 1₂). By contrast, a high neck position entails shorter lever arms (1₃, 1₄) of the same weight forces as before (F₁ and F₂). This reduces the load on the forefeet: F_v2 in comparison to F_v1. The share of body weight carried by the hindlimbs (F_h1 or F_h2, respectively, is total body weight – F_v The elevation of the neck is equivalent to a shift of the CoM in dorsal and caudal direction.</p

A) Schematic neck of a sauropod to show joints (open circles), centers of segment masses (crosses).

<p>B resulting bending moments. Skeletal structures are in black, ligaments are in dark grey and muscular structures are in light grey. The pull of these structures exerts compressive forces in the vertebral column (black). Note that muscular structures of variable lengths are needed to keep the joints in balance against segment weights, in all positions in which the ligaments are not pre-stretched. Stretching of the ligaments leads to forces which make further movement impossible, so setting limits to neck mobility.</p

Direction and loading of spinal processes (neuroapophyses).

<p>The spinal processes of the tail of a sauropod are not exposed to bending if directed along the resultant of all forces acting on them. Instead of the ligamentum superspinale, a longitudinal muscle leads to the same result. If the muscle forces are increased, or the ligaments heavily prestretched, the resultants may well deviate at least temporarily from the spinal processes. In these cases, bending strength is required.</p

Neck of a horse as example of a cursorial mammal.

<p>A) Horse neck plus head in side view. B) Bending moments caused by segment weights in analogy to the sauropod in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0078574#pone-0078574-g004" target="_blank">Figure 4</a>. C) Cross section through a horse neck at the level of cervical 7. This arrangement of structures is highly specialized to sustain the bending moments that occur in the mediosagittal plane and are visible in side view.</p