A resistant method for landmark-based analysis of individual asymmetry in two dimensions

doi:10.1007/s40484-016-0086-x

Quant. Biol.

2016, Vol. 4

Issue (4) : 270-282 https://doi.org/10.1007/s40484-016-0086-x

RESEARCH ARTICLE

A resistant method for landmark-based analysis of individual asymmetry in two dimensions

Sebastián Torcida¹(

),Paula Gonzalez²,Federico Lotto³

¹. Dpto. Matemática-Exactas, UNCPBA. Tandil 7000, Argentina.
². Fac. de Ciencias Naturales y Museo, UNLP-CONICET. La Plata 1900, Argentina
³. Instituto de Veterinaria Ing. Fernando N. Dulout (IGEVET), Facultad de Cs. Veterinarias, UNLP-CONICET. La Plata 1900, Argentina

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Abstract

Background: Symmetry of biological structures can be thought as the repetition of their parts in different positions and orientations. Asymmetry analyses, therefore, focuses on identifying and measuring the location and extent of symmetry departures in such structures. In the context of geometric morphometrics, a key step when studying morphological variation is the estimation of the symmetric shape. The standard procedure uses the least-squares Procrustes superimposition, which by averaging shape differences often underestimates the symmetry departures thus leading to an inaccurate description of the asymmetry pattern. Moreover, the corresponding asymmetry values are neither geometrically intuitive nor visually perceivable.

Methods: In this work, a resistant method for landmark-based asymmetry analysis of individual bilateral symmetric structures in 2D is introduced. A geometrical derivation of this new approach is offered, while its advantages in comparison with the standard method are examined and discussed through a few illustrative examples.

Results: Experimental tests on both artificial and real data show that asymmetry is more effectively measured by using the resistant method because the underlying symmetric shape is better estimated. Therefore, the most asymmetric (respectively symmetric) landmarks are better determined through their large (respectively small) residuals. The percentage of asymmetry that is accounted for by each landmark is an additional revealing measure the new method offers which agrees with the displayed results while helping in their biological interpretation.

Conclusions: The resistant method is a useful exploratory tool for analyzing shape asymmetry in 2D, and it might be the preferable method whenever a non homogeneous deformation of bilateral symmetric structures is possible. By offering a more detailed and rather exhaustive explanation of the asymmetry pattern, this new approach will hopefully contribute to improve the quality of biological or developmental inferences.

Author Summary A resistant method for studying individual shape asymmetry in two dimensions is introduced, which uses the geometry of data to estimate the underlying symmetric shape. Unlike the classical least-squares approach, it is shown show that asymmetry is more accurately measured when a resistant method is used instead; this helps symmetry departures to be more easily understood. The percentage of asymmetry accounted for by each landmark can also be computed in the process, providing an objective basis for a comprehensive characterization of asymmetry. Overall, the resistant method turns out to be a useful exploratory tool whenever a non homogeneous deformation of bilateral symmetric structures is possible.

Keywords resistant procrustes method shape asymmetry matching and object symmetry landmarks

PACS:

Fund:

Corresponding Author(s): Sebastián Torcida

Just Accepted Date: 25 October 2016 Online First Date: 23 November 2016 Issue Date: 01 December 2016

Cite this article:

Sebastiá,n Torcida,Paula Gonzalez, et al. A resistant method for landmark-based analysis of individual asymmetry in two dimensions[J]. Quant. Biol., 2016, 4(4): 270-282.

URL:

https://academic.hep.com.cn/qb/EN/10.1007/s40484-016-0086-x
https://academic.hep.com.cn/qb/EN/Y2016/V4/I4/270

Fig.1 Steps for the resistant analysis of matching symmetry.

(A) The same resistant location center (darkest dots) is computed for both mirror copies. (B) Configurations are translated to place their corresponding resistant center (black dot) at the origin of coordinates. (C) Difference vectors (dotted lines) are first normalized, and their sphmed (arrow) is afterwards computed. The estimated resistant median axis (dashed line) is the line perpendicular to sphmed. (D) Any of the copies (the yellow one in this case) is reflected about the estimated median axis. An additional resistant Procrustes superimposition filters out the eventually remaining differences due to scale and/or orientation.

Fig.2 Matching symmetry: fitted configurations by using the resistant (left) and the LS (right) Procrustes method.

A resistant measure of asymmetry is given by the sum of non-squared Euclidean distances across landmarks, where each landmark contributes the corresponding proportion of exhibited lack of fit. The resistant symmetric shape (not shown) is the row-wise spatial median from the matched configurations.

Tab.1 Matching symmetry geometric example.

Fig.3 Steps for the resistant analysis of object symmetry.

(A) The structure consists of unpaired landmarks 1 to 5, and paired landmarks 6–11, 7–12, 8–13, 9–14, 10–15. (B) A resistant location center (red dot) is computed and the configuration is afterwards translated to place this resistant center at the origin of coordinates. (C) For every pair of unpaired landmarks, the corresponding unit-length direction vectors (arrows) are computed. (D) For every pair of paired landmarks, the corresponding difference vectors (dotted lines) are computed. (E) All paired differences are projected onto every unpaired direction and onto their sphmed direction also (arrows), and the corresponding sum of projection lengths (red lines) is computed. The unpaired or sphmed direction achieving the least sum of projections lengths is the estimated median axis (dashed line). (F) The original configuration (filled dots) is reflected (open dots) about the estimated median axis. The resistant symmetric shape (not shown) is the row-wise spatial median from the original and the reflected configurations.

Fig.4 Object symmetry: original configuration (orange dots) and the estimated symmetric shape (green dots) when the resistant (left) and the LS (right) Procrustes methods are used.

Resistant asymmetry is measured through the sum of non-squared Euclidean distances across landmarks between them; each landmark contributes the proportion of displayed lack of fit in the overall sum.

Tab.2 Object symmetry geometric example.

Fig.5 Simulated biological example of matching symmetry: digitized landmarks from Drosophila melanogaster wings.

Asymmetry between sides was artificially introduced in landmarks 7, 8, 9, 10 and 13.

Fig.6 Resistant vs LS results for the Drosophila melanogaster wings.

One of the configurations (black dots) and the estimated symmetric shape (orange dots) are shown.

Tab.3 Matching symmetry biological example (Drosophila wings).

Fig.7 Simulated biological example of object symmetry: digitized landmarks from a human face.

Landmarks 1 to 6 are unpaired, and paired landmarks are 7-14, 8-15, 9-16, 10-17, 11-18, 12-19 and 13-20. Asymmetry was artificially introduced in unpaired landmarks 4 and 5 and in paired ones 7, 16, 17, 18 and 19.

Fig.8 Resistant vs LS results for the human face.

The original configuration (black dots) and the estimated symmetric shape (orange dots) are shown.

Tab.4 Object symmetry biological example (human face).

1	Debat, V., Milton, C. C., Rutherford, S., Klingenberg, C. P. and Hoffmann, A. A. (2006) Hsp90 and the quantitative variation of wing shape in Drosophila melanogaster. Evolution, 60, 2529–2538 https://doi.org/10.1111/j.0014-3820.2006.tb01887.x pmid: 17263114
2	Gonzalez, P. N., Lotto, F. P. and Hallgrímsson, B. (2014) Canalization and developmental instability of the fetal skull in a mouse model of maternal nutritional stress. Am. J. Phys. Anthropol., 154, 544–553 https://doi.org/10.1002/ajpa.22545 pmid: 24888714
3	Willmore, K. E., Leamy, L. and Hallgrímsson, B. (2006) Effects of developmental and functional interactions on mouse cranial variability through late ontogeny. Evol. Dev., 8, 550–567 https://doi.org/10.1111/j.1525-142X.2006.00127.x pmid: 17073938
4	Palmer, A. R. and Strobeck, C. (1986) Fluctuating asymmetry: measurement, analysis, patterns. Annu. Rev. Ecol. Syst., 17, 391–421 https://doi.org/10.1146/annurev.es.17.110186.002135
5	Adams, D. C., Rohlf, F. J. and Slice, D. E. (2004) Geometric morphometrics: ten years of progress following the ‘revolution’. Ital. J. Zool., 71, 5–16 https://doi.org/10.1080/11250000409356545
6	Adams, D. C., Rohlf, F. J. and Slice, D. E. (2013) A field comes of age: geometric morphometrics in the twenty first century. Hystrix, 24, 7–14
7	Auffray, J. C., Alibert, P., Renaud, S., Orth, A. and Bonhomme, F. (1996). Fluctuating asymmetry in mus musculus sub-specific hybridization: traditional and Procrustes comparative approach. In Advances in Morphometrics. Marcus, L. F. et al. eds., 275–283. New York: Plenum Press
8	Auffray, J. C., Debat, V. and Alibert, P. (1999) Shape asymmetry and developmental stability. In On Growth and Form: Spatio-temporal Pattern Formation in Biology. Chaplain, M. A. J. et al. eds., 309–324. Chichester: Wiley
9	Bookstein, F. L. (1996a) Biometrics, biomathematics and the morphometric synthesis. Bull. Math. Biol., 58, 313–365 https://doi.org/10.1007/BF02458311 pmid: 8713662
10	Bookstein, F. L. (1996b). Combining the tools of geometric morphometrics. In Advances in Morphometrics. L. F. Marcus, L. F. et al. eds., 131–151, New York: Plenum Press
11	Klingenberg, C. P., Barluenga, M. and Meyer, A. (2002) Shape analysis of symmetric structures: quantifying variation among individuals and asymmetry. Evolution, 56, 1909–1920 https://doi.org/10.1111/j.0014-3820.2002.tb00117.x pmid: 12449478
12	Klingenberg, C. P. (2015) Analyzing fluctuating asymmetry with geometric morphometrics: concepts, methods, and applications. Symmetry, 7, 843–934 https://doi.org/10.3390/sym7020843
13	Mardia, K. V., Bookstein, F. L. and Moreton, I. J. (2000) Statistical assessment of bilateral symmetry of shapes. Biometrika, 87, 285–300. https://doi.org/10.1093/biomet/87.2.285
14	Mitteroecker, P. and Gunz, P. (2009) Advances in geometric morphometrics. Evol. Biol., 36, 235–247 https://doi.org/10.1007/s11692-009-9055-x
15	Smith, D. R., Crespi, B. J. and Bookstein, F. L. (1997) Fluctuating asymmetry in the honey bee, Apis mellifera: effects of ploidy and hybridization. J. Evol. Biol., 10, 551–574 https://doi.org/10.1007/s000360050041
16	Catalano, S. A. and Goloboff, P. A. (2012) Simultaneously mapping and superimposing landmark configurations with parsimony as optimality criterion. Syst. Biol., 61, 392–400 https://doi.org/10.1093/sysbio/syr119 pmid: 22213710
17	Richtsmeier, J. T., DeLeon, V. B. and Lele, S. R. (2002) The promise of geometric morphometrics. Am. J. Phys. Anthropol., 119, 63–91 https://doi.org/10.1002/ajpa.10174 pmid: 12653309
18	Theobald, D. L. and Wuttke, D. S. (2006) Empirical Bayes hierarchical models for regularizing maximum likelihood estimation in the matrix Gaussian Procrustes problem. Proc. Natl. Acad. Sci. USA, 103, 18521–18527 https://doi.org/10.1073/pnas.0508445103 pmid: 17130458
19	Van der Linde, K. and Houle, D. (2009) Inferring the nature of allometry from geometric data. Evol. Biol., 36, 311–322 https://doi.org/10.1007/s11692-009-9061-z
20	Siegel, A. F. and Benson, R. H. (1982) A robust comparison of biological shapes. Biometrics, 38, 341–350 https://doi.org/10.2307/2530448 pmid: 6810969
21	Slice, D. E. (1996). Three-dimensional generalized resistant fitting and the comparison of least-squares and resistant fit residuals. In Advances in Morphometrics, Marcus, L. F. et al. eds., 179–199, New York: Plenum Press
22	Torcida, S., Perez, I. and Gonzalez, P. N. (2014) An integrated approach for landmark-based resistant shape analysis in 3D. Evol. Biol., 41, 351–366 https://doi.org/10.1007/s11692-013-9264-1
23	Hampel, F. R., Ronchetti, E. M., Rouseeuw, P. J. and Stahel, W. A. (1986) Robust Statistics: The Approach Based on Influence Functions. New York: Wiley
24	Huber, P. (1981) Robust Statistics. New York: Wiley
25	Siegel, A. F. (1982) Robust regression using repeated medians. Biometrika, 69, 242–244 https://doi.org/10.1093/biomet/69.1.242
26	Klingenberg, C. P. and McIntyre, G. S. (1998) Geometric morphometrics of developmental instability: analyzing patterns of fluctuating asymmetry with Procrustes methods. Evolution, 52, 1363–1375 https://doi.org/10.2307/2411306
27	Cheverud, J. (1995) Morphological integration in the saddleback tamarin (Saguinus fuscicollis) cranium. Am. Nat., 145, 63–89 https://doi.org/10.1086/285728
28	Hallgrímsson, B. and Lieberman, D. E. (2008) Mouse models and the evolutionary developmental biology of the skull. Integr. Comp. Biol., 48, 373–384 https://doi.org/10.1093/icb/icn076 pmid: 21669799
29	Zelditch, M. L., Swiderski, D. L., Sheets, H. D. and Fink, W. L. (2004) Geometric Morphometric for Biologists. London: Academic Press
30	Slice, D. E. (2001) Landmark coordinates aligned by procrustes analysis do not lie in Kendall’s shape space. Syst. Biol., 50, 141–149. https://doi.org/10.1080/10635150119110 pmid: 12116591
31	Gower, J. C. (1975) Generalized procrustes analysis. Psychometrika, 40, 33–51 https://doi.org/10.1007/BF02291478
32	Rohlf, F. J. and Slice, D. E. (1990) Extensions of the Procrustes method for the optimal superimposition of landmarks. Syst. Zool., 39, 40–59 https://doi.org/10.2307/2992207
33	Bookstein, F. L. (1991) Morphometric Tools for Landmark Data: Geometry and Biology. New York: Cambridge University Press
34	Weiszfeld, E. (1937) On the point for which the sum of the distances to n given points is minimum. Tohoku Math. J., 43, 355–386
35	Xue, G. L. (1994) A globally convergent algorithm for facility location on a sphere. Comput. Math. Appl., 27, 37–50 https://doi.org/10.1016/0898-1221(94)90109-0

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