Collaborator: Peter C. Wainwright
Biomechanical models are important tools for understanding the functional implications of morphological evolution. It is therefore vital to employ effective methods for capturing the diversity of motions that the underlying mechanical systems generate. For this research, we focused on a commonly used model for studying the mechanics of oral jaws in perciform fishes, the fourbar linkage. The oral jaw fourbar linkage is a rigid mechanism defined by skeletal features of the feeding apparatus and is responsible for generating anterior protrusion of the mouth during suction-based feeding. We used geometric morphometrics to study the kinematic properties of fourbar linkages in the cichlids of Madagascar. When any component of the fourbar linkage undergoes rotational change, the other links move in a deterministic manner such that shape change across the entire linkage configuration can be captured as a trajectory through morphospace (Fig. 1A). The characteristics of these trajectories, like their lengths and shapes, provide information about the nature of motion diversity associated with fourbar linkages (Fig. 1B).
Using shape trajectories captured in simulated fourbar linkage motions, we computed two functional metrics. The first, kinesis, is the amount of shape change that the linkage undergoes given 30 degrees of input rotation of the lower jaw. It is calculated as the total length of the shape trajectory (Fig. 1B). The other is kinematic asynchrony (KA), which is the relative degree of nonlinearity of the trajectory. It is measured as the maximum deviation (Procrustes distance) of a shape along a motion trajectory to a corresponding shape along a hypothetical linear baseline between start and end configurations, standardized by the length of the linear baseline (Fig. 1B). KA captures the degree to which different mobile components of a functional system (here, landmarks or linkage joints) are differentially activated over time during a motion. For comparison, shape change along a linear trajectory is characterized by synchronous movements of components, where all landmarks accelerate proportionately to each other and each maintains a single direction of change (Fig. 2). While we have long known that different parts of functional systems often vary in the timing of peak activation, but the current methods highlight how this feature of motion diversity contributes to overall kinesis.
In addition to calculating functional metrics for fourbar linkages of actual species, we can also estimate the linkage shapes and their functions across the morphospace defined by those species (Fig. 3). This form-function landscape provides context to the patterns of variation observed in organisms. For example, we can see that for kinesis (Fig. 3A), etropline cichlids (white dots) vary in morphology in a direction that minimizes variation in oral jaw mobility relative to ptychochromines (black dots). Additionally, the results suggest that KA is similar but not identical to another common functional metric for linkage models, kinematic transmission (KT). In oral jaw fourbars, KT is measured as the ratio of output rotation of the maxillary link to input rotation of the lower jaw link, and describes how motion is transferred from one part of the linkage to another. An advantage to calculating functional metrics from shape trajectories, as opposed to KT, is that the they are applied to the entire mechanical system, versus pairs of components at a time. Additionally, the same methods can be applied to more complicated biomechanical models as well as live organisms (Martinez et al. 2018), allowing researchers to use very different types of data to produce results that can be interpreted in the same way.
Martinez CM & Wainwright PC. In Review. Extending the Geometric Approach for Studying Biomechanical Motions. Integrative & Comparative Biology.
Martinez CM, McGee MD, Borstein SR & Wainwright PC. 2018. Feeding ecology underlies the evolution of cichlid jaw mobility. Evolution. 72 (8), 1645-1655.