Evolutionary Biology

, Volume 39, Issue 3, pp 419–439 | Cite as

The Measurement of Local Variation in Shape

  • Eladio J. Márquez
  • Ryan Cabeen
  • Roger P. Woods
  • David Houle
Tools and Techniques

Abstract

Geometric morphometrics comprises tools for measuring and analyzing shape as captured by an entire set of landmark configurations. Many interesting questions in evolutionary, genetic, and developmental research, however, are only meaningful at a local level, where a focus on “parts” or “traits” takes priority over properties of wholes. To study variational properties of such traits, current approaches partition configurations into subsets of landmarks which are then studied separately. This approach is unable to fully capture both variational and spatial characteristics of these subsets because interpretability of shape differences is context-dependent. Landmarks omitted from a partition usually contain information about that partition’s shape. We present an interpolation-based approach that can be used to model shape differences at a local, infinitesimal level as a function of information available globally. This approach belongs in a large family of methods that see shape differences as continuous “fields” spanning an entire structure, for which landmarks serve as reference parameters rather than as data. We show, via analyses of simulated and real data, how interpolation models provide a more accurate representation of regional shapes than partitioned data. A key difference of this interpolation approach from current morphometric practice is that one must assume an explicit interpolation model, which in turn implies a particular kind of behavior of the regions between landmarks. This choice presents novel methodological challenges, but also an opportunity to incorporate and test biomechanical models that have sought to explain tissue-level processes underlying the generation of morphological shape.

Keywords

Geometric morphometrics Thin-plate splines Shape variables Interpolation Local shape variation Modularity Biomechanical models 

Notes

Acknowledgments

This work was funded by the National Institutes of Health through the NIH Roadmap for Medical Research, Grant U54 RR021813 and through National Science Foundation grant DEB-0950002. Information on the National Centers for Biomedical Computing can be obtained from http://nihroadmap.nih.gov/bioinformatics. We thank Thomas Hansen for his valuable insights on uses of interpolation methods, and J. Kent and three anonymous reviewers for insightful comments. Analyses and simulations were based on code written on SAS, Matlab, Java, C++, and Python. Code used for simulations and in the computation of Jacobians for Drosophila wing data is available upon request from E.M. and R.C., respectively. In addition, a user-friendly Matlab-based standalone package to carry out most of the analyses and to generate all of the graphic outputs shown in this paper has been made available at http://bio.fsu.edu/~dhoule/Software/.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Eladio J. Márquez
    • 1
  • Ryan Cabeen
    • 2
  • Roger P. Woods
    • 2
  • David Houle
    • 1
  1. 1.Department of Biological ScienceFlorida State UniversityTallahasseeUSA
  2. 2.Department of NeurologyUCLA School of MedicineLos AngelesUSA

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