GEOL 331 Principles of Paleontology

Fall 2016

Describing Morphology

Acknowledgment: Much of the following material is adapted from David Polley's G562 Geometric Morphometrics course material to which serious students are directed.

We have seen the Raup 1966 attempt to reduce morphological variation to a small number of parameters. In all but the simplest systems, such a method greatly oversimplifies morphology. Nevertheless, the need to describe morphology with quantitative rigor is fundamental to descriptive paleontology, and to the testing of hypotheses of functional morphology.

In this lecture we consider morphology and function separately.

Morphology

Morphometrics: The quantitative study of shape. Fundamental to the description of taxa is the need to distinguish them from one another by repeatable means and to describe variation in shape, separate from other characteristics such as size. There are three general approaches to the generation of morphometric data that can be subjected to statistical analyses.

We care because it is only through this means that we can effectively explore theoretical morphospace, addressing issues including:

• Alpha taxonomy: Assessing the grouping of individuals into species, sexual morphs, and growth stages
• Issues of biomechanics: including allometric scaling and the limits and constraints of shape on evolution.
• The evolution of ecospace: including the occupation and evacuation of ecological guilds - groups with similar life styles, regardless of phylogeny over evolutionary time.

Traditional morphometrics: Measurements that have become standardized for the description of morphology through tradition, usually owing to their lack of ambiguity. Naturalists have been using calipers to obtain quantitative information about morphology for centuries. That information has been fed into multivariate statistical analyses for a full century. These begin to enable us to describe distributions of morphological variation.

Possible statistical analyses include:

From Wikipedia
• Simple linear regression: A cluster of bivariate data points are reduced to a line that minimizes the residals - distances from the line to the data points - using the least squares method, which minimizes the square root of the sum of their squares. The regression line represents an estimate of the relationship between the variables. The more strongly correlated the variables, the closer the cluster of points will fall to the regression line.
  
  
  From Wikipedia
• Principal Component Analysis: A cluster of data points (bivariate or multivariate) are mapped onto a new cartesian coordinate system with a new set of axes - principal components - that are mutually perpendicular and separate non-correlated variation. In them:
  • The first principal component represents the primary source of variation, the second, the secondary, etc..
  • There can be as many principal components as there are variables.
  • The principal components, being mutually perpendicular, are not correlated with one another.
  Thus, PCA is useful in identifying separate sources of variation in a population and quantifying their relative contribution to overall variation. Of course, data can be displayed using any of these axes that the researcher deems appropriate.
  
  
  
• Discriminant function Analysis: Suppose two distinct groups are known a priori (for example, herbivores and carnivores), and two variables, x and y, are measured and plotted. We see that the populations form two clusters, but neither variable, x nor y, does a good job of separating them by itself. A discriminant function (blue line at right) can be identified, however, that maximizes the distinction between them. Were we to introduce data for a new specimen whose group membership is unknown, we could estimate it based on where it fell on the discriminant function.

  This may be trivial for a bivariate plot, but discriminant function analysis also works for multivariate analyses in many dimensions, where data cannot be as easily visualized.
  

  
  

But what do the data actually mean? In traditional morphometrics, data are measurements of the distances between points on a specimen. These points may be:

• Landmarks: Discrete points marked by homologous features (right - red) like:
  • Foramina
  • Intersections of sutures in bones
  • Branching points of veins
  • Other similar points
• Semilandmarks: Arbitrary points on the exterior margin describe shape but may be non-homologous (right - blue) like:
  • points of maximum width
  • the tips of snouts
  • etc.

These measurements are indirectly related to shape, but are, in fact measures of distance. Thus, they are useful in examining scaling effects, but in extracting them we disregard direct information on shape. Indeed frequently, the first principal component extracted from such data is usually dominated by differences in size and correlated variables.

D'Arcy Thompson: (1860 - 1948) Was the first pioneer in the investigation of the relationship between growth, evolution, and geometric form. His classic On Growth and Form identifies such famous examples as the relationship between the Fibonaci sequence and spiral patterns in plant growth. In fact, he was preoccupied by the constraints imposed by the interaction of simple transformational processes and specific initial conditions.

Thompson was fond of expressing morphological variation in terms of "transformations" - by which the shape of one species could be transformed into that of another by the application of regular simple transformations, reflected as deformed grids. (E.g. puffer and mola, right.) Although clever and not lacking merit, these reflected a level of subjectivity. Indeed, the technique was pioneered by the Renaissance artist Albrecht Dürer. They pointed out the need for an algorithmic approach to the quantitative analysis of shape, however.

This was ultimately supplied during the 1990s by Fred Bookstein, F. James Rohlf, and colleagues. (See review by Adams et al., 2013.) Here is a synopsis of what has emerged.

Landmark-based analysis: Dismisses the measurements of traditional morphometrics and focuses on the distribution of the landmarks - homologous identifiable points in sets of comparable samples. Landmark positions are first recorded as cartesian coordinates, thus preserving information about shape.

Example: For illustration, we show three specimens with different shapes. In practice, a normal data set would contain more. Constraints:

• The data must reflect shape adequately.
• Comparable landmarks must be present on all specimens.
• The number of specimens required depends on the size of the difference you seek to study compared to the amount of variation in your group.

Step 1: Data acquisition. Can be through:

• Digital cameras
• Calipers

Step 2: Procrustes superimposition. A series of steps in which size, positional, and orientation data are removed, leaving only shape.

• Centroids calculated for all shapes
• Size and orientation normalized using least squares concept.

Step 3: Principal Component Analysis. With size and orientation information removed, landmarks can now be treated as variables in PCA. The results map non-correlated sources of variation in shape without respect to size. This can be explored graphically with reference to the original data (right) or through thin-plate spline grids.

Outline-based analysis: An alternate approach involves describing the outline of an object mathematically by the fitting of coefficients of mathematical functions to a regular array of preset semilandmarks around the outline. In this case, there is no expectation that the points represent homologous features.

Eigenshape analysis: A common method of mathematical description is eigenshape analysis, in which a preset number of regularly spaced semilandmarks approximate the object's shape. For each semilandmark, the deviation of the angle of the line connecting it to the next from a circle centered on the centroid is recorded for each semilandmark, becoming a variable in a PCA, whose results can be expressed graphically.

For closed curves, Elliptical fourier analysis: (Not pictured) In which the shape of the outline is approximated by the sum of the minimum number of ellipses needed to mimic its shape. This only works on outlines that are consistently convex.

Morphometrics in three dimensions: Most of the techniques discussed here can in principle be adapted for use with three-dimensional objects, although logistics become more complex.

Data acquisition: Depending on the size of the subject:

• Reflex microscopes
• Microscribe robotic arms
• laser scanners
• CT scanners

Function

Finite Element Analysis: From a three dimensional landmark model, it is a sort step to the use of three dimensional models in the actual testing of functional hypotheses through finite element analysis. (Link to video) This method was developed independently by several researchers in North America, Europe, and China during the mid-20th century as a practical approach to problems of civil and mechanical engineering requiring the simultaneous solution of many partial differential equations.

The steps:

• Complex shapes like bridges, for which the equations are known but solutions are impossible, are broken down in to a series of finite elements - simple shapes for which the equations can be solved. Together, these are the finite element mesh. The composite solution of equations for each element approximates a solution for the complex object being modelled.
  • Finite elements can be on dimensional rods, two dimensional polygons, or simple three dimensional polyhedrons.
  • The more elements, the more computations are required but the smaller the error. Thus a compromise is sought. An advantage of the method is that the mesh can have more detail in important regions and less detail in unimportant ones.
• Nodes: Corners and surfaces of finite elements, where they interface with adjacent elements.
• The displacement of nodes in a given element serves as the basis for the calculation of system equations for the entire element, which are represented as a matrix
• Because adjacent elements share nodes, their matrices overlap. Through a series of matrix calculations, stress and strain experienced by each element can be calculated and mapped.
• These values can then be fed into secondary iterations of the process to simulate sequences of events such as fluid flow.

Applications of FEA to paleontology:

Hyopthesis testing: Recall that among the few iron-clad tests of function in paleontology are assessments of whether a proposed function requires a biological material to exceed its known strength. In all but the simplest cases, such analysis would be impossible but for FEA.

As luck would have it, the rise of FEA has paralleled the rise of methods for capturing three-dimensional information about morphology, especially:

• Three-D laser scanners
• CT scanners (E.G. Thrinaxodon, Rowe et al., 1993.)

Consequently, FEA has become a promising technique in functional morphology. Some noteworthy examples include:

• Rayfield et al., 2001 on the distribution of bite forces in Allosaurus (see above). Rayfield et al. demonstrated that the skull of Allosaurus was over-engineered with respect to bite forces and proposed the "hatchet-head" predation mode in which the creature attacked its prey by accelerating its teeth into it.

• Tseng, 2013 on the convergent evolution of borophagine canids (bone-cracking dogs) and Hyenas.

• Wroe et al., 2013 on the biomechanics of bite forces in the sabre-toothed cat Smilodon fatalis and the sabre-toothed marsupial Thylacosmilus atrox, in which the latter is deduced to have inserted its canines primarily using the force generated by its neck muscles.

Caveats

Correlation studies: Biologists are fond of performing statistical studies, looking for meaningful correlations between measurable aspects of different parts of animals' anatomies. For instance, one might look at the ratio of the length and depth of birds' beaks vs. the length of their tarsals to see if larger birds have proportionately deeper beaks. Measurements of this sort can be subjected to the full range of statistical analyses. A fundamental assumption of such studies is that all observations be made from independent members of the same underlying population.

However, when samples are being taken from groups of populations nested in a heirarchical phylogeny, this assumption is violated. The example at right shows of how misleading this can be. Imagine a simple phylogeny with two sister clades, one black, the other red. When we make a bivariate plot of data taken from them, there appears to be a strong statistical relationship. But in fact, when we compare values only within each clade, we see very low correlation.

Fortunately, Joe Felsenstein of the University of Washington provided the solution in Felsenstein, 1985: a statistical technique for adjusting data to account for the phylogeny of the taxa sampled, called phylogenetically independent contrasts (PIC). This method is based on the concept that although the taxa may be non-independent, the differences between measured values in them are independent. Statistical correlation techniques are, therefore, applied to pairwise contrasts in measurements from sister taxa. By applying it, meaningful correlation studies can be performed. Without it, they would be meaningless or misleading. Applying this method absolutely requires a known phylogeny. If hypotheses of phylogeny change, the results of correlation studies based on them must be revised, too.