Morphospace: Measurement, Modeling, Mathematics, and Meaning

N. Khiripet¹ ; R. Viruchpintu¹ ; J. Maneewattanapluk¹ ; J. Spangenberg² ; J.R. Jungck³

¹ Bioinformatics Unit, Knowledge Elicitation and Archiving Laboratory (KEA) NECTEC (National Electronics and Computer Technology Center), Thailand Science Park 112 Phahonyothin Road, Klong 1, Klong Luang, Pathumthani 12120, Thailand
² Department of Biology, Washington State University, Pullman, WA 99164
³ Department of Biology, Beloit College, 700 College Street, Beloit, WI 53511

Mathematical modelling of natural phenomena, Tome 6 (2011) no. 2, pp. 54-81.

Résumé

Artists have long recognized that trees are self-similar across enormous differences in magnitudes; i.e., they share a common fractal structure - a trunk subdivides into branches which subdivide into more branches which eventually terminate in leaves, flowers, fruits, etc. Artistid Lindenmayer (1971, 1975, 1989, 1990) invented a mathematics based on graph grammar rewriting systems to describe such iteratively branching structures; these were named in honor of him and are referred to as L-systems. With the advent of fractals into computer graphics, numerous artists have similarly produced a wide variety of software packages to illustrate the beauty of fractal/L-system generated plants. Some tree visualizations such as L-Peach (Allen , 2005) do depend very explicitly upon a complex set of precise measurements of a single species of tree. Nonetheless, we felt that there is a need to build a package that allowed scientists (and students) to collect data from actual specimens in the field or laboratory, insert these measurements into an L-system package, and then visually compare actual trees to the computer generated image with such specimens. Furthermore, the effect of variance in parameters helps users evaluate the developmental plasticity both within and between species and varieties. We have developed 3D FractaL Tree (the L is capitalized in honor of Lindemayer) to generate trees based upon measurement of (1) relative lengths of two successive segments averaged over several iterations, (2) the angle theta between bifurcating limbs at successive joints, (3) the number of steps in branching that one must follow to find a branch extending at the same angle as the first one under consideration to determine the phyllotactic angle phi, (4) the average of the summed areas (determined from measurement of diameters) of bifurcations compared to the trunk to determine whether area of flow is preserved (and to consider Poiseuille’s/Murray’s law of laminar flow in a fractal network), (5) the total number of iterative branching from the base to the tip of tree averaged over several counts based on following out different major limbs, (6) an editable L-system rule chosen from a library of branching patterns that roughly correspond to a specimen under consideration, and (7) a degree of stochasticity applied to the above rules to represent some variation over the course of a lifetime. Of course, turned upside down, the computer imagery could be used to represent root structure instead of above ground growth or the bronchial system of a lung, for example. The measurements are recorded and analyzed in a series of worksheets in Microsoft Excel and the results are entered into the graphics engine in a Java application. 3D FractaL Tree produces a rotatable three-dimensional image of the tree which is helpful for examining such characters as self-avoidance (entanglement and breakage), reception of and penetration of sunlight, distances that small herbivores (such as caterpillars) would have to traverse to go from one tip to another, allometric relationships between the convex hull of the crown (as perceived in a top-down projection of the tree) and the trunk’s diameter, and convex hull of the volume distribution of biomass on different subsections of a tree which have been discussed in the Adaptive Geometry of Trees (Horn, 1971) and subsequent research for the past four decades. Besides being able to rotate the three dimensional tree in the x-y, y-z, and x-z planes as well as zoom-in and zoom-out, three different representations are available in 3D FractaL Tree images: wire frame, solid, and transparent. Easy options for editing L-system rules and saving and exporting images are included. 3D FractaL-Tree is published with a Creative Commons license so that it is freely available for downloading, use, and extending with attribution from our Biological ESTEEM Project (http://bioquest.org/esteem).

Détail

DOI : 10.1051/mmnp/20116202

Bibliographie

[1] M. T. Allen, P. Prusinkiewicz, T. M. Dejong Using L-systems for modeling source-sink interactions, architecture and physiology of growing tree: The L-Peach model New Phytologist. 2005 869 880

[2] R. Aratsui Leonardo was wise: Trees conserve cross-sectional area despite vessel structure J. Young Investigators 1998 1 23

[3] J. K. Bailey, R. K. Bangert, J. A. Schweitzer, R. T. Trote. Iii, S. M. Shuster, T. G. Whitman Fractal geometry is heritable in trees Evolution 2004 2100 2102

[4] D. Barthélémy, Y. Caraglio Plant architecture: A dynamic, multilevel and comprehensive approach to plant form, structure and ontogeny Annals of Botany 2007 375 407

[5] F. S. Berezovskava, G. P. Karev, O. S. Kisliuk., R. G. Khlebopros, Y. L. Tsel’Niker A fractal approach to computer-analytical modeling of tree crowns Trees 1997 323 327

[6] N. Bessonov, V. Volpert. Dynamic models of plant growth. Mathematics and Mathematical Modelling. Editions Publibook Universite, Paris, France, 2008.

[7] L. Breiman. Reflections after refereeing papers for NIPS. In David Wolpert, Ed., The Mathematics of Generalization. Santa Fe Institute Studies in the Sciences of Complexity Volume XX. Addison-Wesley: Reading, MA, 1995.

[8] J. L. Casti. Would-Be worlds: How simulation is changing the frontiers of science. John Wiley and Sons: New York, NY, 1997.

[9] J. E. Cohen Mathematics is biology’s next microscope, only better Public Library of Science Biology 2004 2017 2023

[10] O. Deussen, B. Lintermann. Digital design of nature: Computer generated plants and organics. New York: Springer-Verlag, 2005.

[11] A. E. Dimond Pressure and flow relations in vascular bundles of the tomato plant Plant Physiology 1966 119 131

[12] J. Elkins. The object stares back: On the nature of seeing. Harcourt Brace and Company: San Diego, CA, 1996.

[13] B. J. Enquist, K. J. Niklas Global allocation rules for patterns of biomass partitioning in seed plants Science 2002 1517 1520

[14]

[15] FSPM07. Fifth international workshop on functional structural plant models. Napier, New Zealand, 4–9 November, 2007.

[16] J. Fisher, H. Honda Branch geometry and effective leaf area: A study of Terminalia-branching pattern: Theoretical ideas American J. Botany 1979 633 644

[17] C. Godin, O. Puech, F. Boudon, H. Sinoquet. Space occupation by tree crowns obey fractal laws: Evidence from 3D digitized plants. In Godin, C. Hanan, J. Kurth, W. Lacointe, A. Takenaka, A. Prusinkiewicz, P. DeJong, T. M. Beveridge, C. Andrieu, B., Eds. Proceedings of Proceedings of the 4th International Workshop on Functional-Structural Plant Models. Montpellier, 2004, 79–83. Proceedings of the 4th International Workshop on Functional-Structural Plant Models, 7-11/06/2004, Montpellier.

[18] C. Godin, E. Costes, H. Sinoquet A method for describing plant architecture which integrates topology and geometry Annals of Botany 2004 343 357

[19] Y. Grossman, T. De Jong, Peach. In Jungck , Eds., The BioQUEST Library, Academic Press: San Diego, 2003.

[20] M. Hiratsuka, T. Toma, M. Yamada, I. Heriansyah, Y. Morikawa. A general allometric equation for estimating biomass in Acacia mangium plantations. In Proceedings of the 2003 International Conference on Tropical Forests and Climate Change, University of the Philippines Los Banos, Laguna, Philippines 2003, 212–218.

[21] P. Hogeweg, B. Hesper A model study on biomorphological description Pattern Recognition 1974 165 179

[22] H. Honda Description of the form of trees by the parameters of a tree-like body: Effects of the branching angle and branch length on the shape of the tree-like body J. Theoretical Biology 1971 331 338

[23] H. S. Horn. Adaptive geometry of trees. Princeton University Press: Princeton, NY, 1971.

[24] J. R. Jungck Ten equations that changed biology Bioscene: Journal of College Biology Teaching 1997 11 36

[25] J. R. Jungck, E. D. Stanley, S. J. Everse, V. Vaughan, Eds. The BioQUEST Library. Academic Press: North Reading, MA, 2003.

[26] J. R. Jungck The biological ESTEEM collection: Excel simulations and tools for exploratory, experiential mathematics BioQUEST Notes 2005 1 2

[27] J. R. Jungck. Challenges, connections, complexities: Educating for collaboration. Chapter in Lynn Steen, Ed., Math Bio2010: Linking Undergraduate Disciplines. The Mathematical Association of America: Washington, DC., 2005, 1–12.

[28] J. R. Jungck. Fostering figuring and fascination: Engaging learners through alternative aesthetics. In J. R. Jungck, Er. Mazur, Wi. Schmidt, T. Schuller, Eds., Criticism of Contemporary Issues: Education. Serralves Museum of Art, Fundacao Serralves: Porto, Portugal, 2008, 19–54 and 93–126.

[29] C. E. Knapp. Review of Last child in the woods: Saving our children from nature-deficit disorder (Revised edition), by Richard Louv. 2008. Chapel Hill, NC: Algonquin Books. 390 pages. Paperback. $14.95. ISBN13: 978-1-56512-605-3. Journal of Environmental Education, 40 2009, No. 2, 63–64.

[30] A. Lindenmayer Developmental systems without cellular interaction, their languages and grammars J. Theoretical Biology 1971 455 484

[31] A. Lindenmayer. Developmental systems and languages in their biological context. In G. T. Herman G. Rozenberg, Eds., Developmental systems and languages. Amsterdam: North-Holland Publishing, 1975, 1–40. 0-7204-2806-8

[32] A. Lindenmayer. Growing fractals and plants. In Heinz-Otto Peitgen, Hartmut Jürgens and Dietmar Saupe, Eds. Chaos and Fractals, Second Edition. New Frontiers of Science. Springer: New York, 2004, 329–376. ISBN 978-0-387-20229-7

[33] R. Louv A brief history of the children J. Sci. Educ. Technol 2008 217 218

[34] B. Mandelbrot. Fractals: Form, chance and dimension. W. H. Freeman and Co.: San Francisco, 1977.

[35] K. A. Mcculloh, J. S. Sperry, R. F. Adler Water transport in plants obey Murray’s law Nature 2003

[36] K. A. Mcculloh, J. S. Sperry, R. F. Adler Murray’s law and the hydraulic vs mechanical functioning of wood Functional Ecology 2004

[37] M. Mesterton-Gibbons, M. J. Childress Constraints on reciprocity for non-sessile organisms Bulletin of Mathematical Biology 1996 861 875

[38] National Research Council. Bio 2010: Transforming undergraduate education for future research biology. National Academies Press: Washington, DC, 2003.

[39] K. J. Niklas. Computer simulations of branching-patterns and their implications on the evolution of plants. In L. J. Gross, R. M. Miura, Eds. Some Mathematical Questions in Biology: Plant Biology, Lectures on Mathematics in the Life Sciences, 18, 1986, 1–50 (American Mathematical Society).

[40] K. J. Niklas Evolutionary walks through a land plant morphoscape J. Experimental Botany 1999 39 52

[41] K. J. Niklas, E. D. Cobb Evidence for “diminishing returns” from the scaling of stem diameter and specific leaf area Am. J. Botany 2008 549 557

[42] K. Ogle, S. W. Pacala A modeling framework for inferring tree growth and allocation from physiological, morphological and allometric traits Tree Physiol. 2009 587 605

[43] R. W. Pearcy, H. Muraoka, F. Valladares Crown architecture in sun and shade environments: Assessing function and trade-offs with a three-dimensional simulation model New Phytologist 2005 791 800

[44] C. A. Price, B. J. Enquist, V. M. Savage A general model for allometric covariation in botanical form and function PNAS 2007 13204 13209

[45] P. Przemyslaw, J. Hanan. Lindenmayer systems, fractals, and plants. Lecture Notes in Biomathematics, volume 79. Springer-Verlag, New York, 1989.

[46] P. Prusinkiewicz, A. Lindenmayer. The algorithmic beauty of plants. New York: Springer-Verlag, 1990. (Available on-line in high quality pdf at (http://algorithmicbotany.org/papers/#abop)).

[47] P. Prusinkiewicz, Y. Erasmus, B. Lane, L. D. Harder, E. Coen Evolution and development of inflorescence architectures Science 2007 1452 1456

[48] D. M. Raup Computer as aid in describing form in gastropod shells Science 1962 150 152

[49] P. de Reffye. Computer simulation of plant growth. In Clifford A. Pickover and Stuart K. Tewksbury, Eds. Frontiers of scientific visualization. John Wiley Sons, Inc.: New York, NY, USA, 1994, 145–179.

[50] D. Robinson Implications of a large global root biomass for carbon sink estimates and for soil carbon dynamics Proc. R. Soc. B 2007 2753 2759

[51] A. Szalay, J. Gray 2020 computing: Science in an exponential world Nature 2006 413 414

[52] R. Sattler, R. Rutishauser The fundamental relevance of morphology and morphogenesis to plant research Annals of Botany 1997 571 582

[53] Seits, I.S. Antonova. Applying multidimensional statistics to tree architecture analysis. In Godin, C. Hanan, J. Kurth, W. Lacointe, A. Takenaka, A. Prusinkiewicz, P. DeJong, T. M.Beveridge, C. Andrieu, B., Eds. Proceedings of Proceedings of the 4th International Workshop on Functional-Structural Plant Models. Montpellier, 2004, 70–74. Proceedings of the 4th International Workshop on Functional-Structural Plant Models, 7–11/06/2004, Montpellier.

[54] I. Shlyakhter, M. Rozenoer, J. Dorsey, S. Teller. Reconstructing 3D tree models from instrumented photographs. IEEE Computer Graphics and Animations, xx, 2001, 1–9.

[55] S. Wegrzyn, J.-C. Gille, P. Vidal. Developmental systems: At the crossroads of system theory, computer science, and genetic engineering. Springer-Verlag: New York, 1990.

[56] A. Weisstein, Ed. The Biological ESTEEM project (Excel simulations and tools for exploratory, experiential mathematics), 2009. ( http//bioquest.org/esteem >).

[57] D. Willis, The sand dollar and the slide rule. Addison-Wesley Publishing Co.: North Reading, MA, 232 pp, 1995. ISBN 0-201-63275-6 $23 ($27 postpaid) from

[58] W. Wimsatt. False models as means to truer theories. Chapter 6 in his book:Re-engineering philosophy for limited beings: Piecewise approximations to reality. Harvard University Press: Cambridge, MA, 2007.

[59] M. S. Zens, C. O. Webb Sizing up the shape of life Science 2002