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).


Figure 1.  (A) Motion generated from 30 degrees of lower jaw rotation in the oral jaw fourbar linkage of Paratilapia polleni is summarized on principle components (PCs) 1 and 2. As the motion progresses from start to end configurations (filled dots from left to right), shape change in the fourbar linkage traces a trajectory through morphospace (solid yellow line). Deformation grids of select shapes relative to mean shape are included for visualization along the trajectory. Shapes estimated along a linear trajectory (open dots on dotted yellow line) display the minimum amount of shape change possible between start and end shapes. (B) Functional metrics were estimated from fourbar trajectories. The total amount of shape change, kinesis, was the sum of Procrustes distances between motion shapes (Dto D9). Trajectory nonlinearity, or kinematic asynchrony, was calculated as the maximum Procrustes distance of motion trajectory points from linear (Nto N8), standardized by the linear distance between start and end shapes (L).


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.


Figure 2. (A) Simulated movement of a fourbar linkage with 30 degrees of lower jaw input rotation (left, filled circles) and associated motion along a hypothetical linear trajectory (right, open circles). For the same start and end shapes as the actual motion, landmark movements in a linear motion follow paths that minimize their total distances traveled, but do not represent true linkage movements (link lengths change during the motion). The motion shown is for the linkage shape with maximum kinematic asynchrony (KA) across the Malagasy cichlid morphospace. The two immobile landmarks (LMs) are shown in purple, LM 2 is blue, and LM 3 is green. Red arrows indicate the primary directions of landmark movement (note, LMs for linear trajectories both move in straight lines). (B) Cumulative distances traveled by mobile landmarks at each motion stage are plotted to show how movement is accumulated in the actual fourbar motion (filled dots and solid line) versus the linear version (open dots and dotted line). Colors of axis labels correspond to landmark colors. For the actual motion, LM 2 moves slow at first relative to LM 3, but then accelerates. These asynchronous movements generate nonlinearity of the shape trajectory compared to the linear motion, where both LMs start slower and accelerate proportionately toward the end. (C) For each mobile landmark, slopes are measured between subsequent motion stages, representing the direction of landmark movement. Color and fill patterns correspond to landmarks. Directions of movement vary for the actual fourbar motion (filled circles), but not for the linear motion (open circles).


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.


PC 1&2, fourbar shape, KT 2
Figure 3. Form-function landscapes for three functional traits of fourbar linkages, kinesis (A), kinematic asynchrony (C), and kinematic transmission (E). Each landscape is estimated across the oral jaw fourbar morphospace for Malagasy cichlids, represented here by PCs 1 and 2 (94.6% of total shape variation). Observations are colored by cichlid subfamily, which include Ptychochrominae (black dots) and Etroplinae (white dots). Trait values are represented by plot background color and dotted contour lines show directions of equivalent function across the space. Fourbar shapes at PC extremes are provided on the plots.


Relevant Publications:

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.