Networks with Side Branching in Biology

There are many examples of branching networks in biology. Examples include the structure of plants and trees as well as cardiovascular and bronchial systems. In many cases these networks are self-similar and exhibit fractal scaling. In this paper we introduce the Tokunaga taxonomy for the side branching of networks and his parameterization of self-similar side-branching. We introduce several examples of deterministic branching networks and show that constructions with the same fractal dimension can have different side-branching parameters. As an example of stochastic-branching in biology we consider the vein structure of a leaf. We show that the vein structure of the leaf and river networks have nearly identical side branching statistics. We introduce diffusion limited aggregation (DLA) clusters. These clusters also exhibit Tokunaga side-branching statistics. We consider several alternative explanations for why leaves, river networks, and DLA clusters have similar side-branching statistics. We also consider the allometric scaling relation between metabolic rate and the mass of species in terms of a cardiovascular system with Tokunaga statistics. We find reasonably good agreement between the observed scaling exponent, 10.75, and our model for a range of values of the fractal dimension of the network and the blood flow resistance parameter. 7 1998 Academic Press

[1]  L. Sander.  Diffusion-limited aggregation: A kinetic critical phenomenon? , 2000 .

[2]  William I. Newman,et al.  Fractal Trees with Side Branching , 1997 .

[3]  James H. Brown,et al.  A General Model for the Origin of Allometric Scaling Laws in Biology , 1997, Science.

[4]  Sinclair,et al.  Mechanism for global optimization of river networks from local erosion rules. , 1996, Physical review letters.

[5]  S. Peckham.  New Results for Self‐Similar Trees with Applications to River Networks , 1995 .

[6]  K. Niklas.  Size‐dependent variations in plant growth rates and the “¾‐power rule” , 1994 .

[7]  D. Turcotte,et al.  A diffusion-limited aggregation model for the evolution of drainage networks , 1993 .

[8]  Andrea Rinaldo,et al.  Optimal Channel Networks - a Framework for the Study of River Basin Morphology , 1993 .

[9]  T. Wonnacott,et al.  Relation between diameter and flow in major branches of the arch of the aorta. , 1992, Journal of biomechanics.

[10]  D. Turcotte.  Fractals and Chaos in Geology and Geophysics , 1992 .

[11]  E. Weibel.  Fractal geometry: a design principle for living organisms. , 1991, The American journal of physiology.

[12]  West,et al.  Complex fractal dimension of the bronchial tree. , 1991, Physical review letters.

[13]  Paul Meakin,et al.  Fractal aggregates in geophysics , 1991 .

[14]  B. Zeide,et al.  Fractal dimensions of tree crowns in three loblolly pine plantations of coastal South Carolina , 1991 .

[15]  M. Labarbera.  Principles of design of fluid transport systems in zoology. , 1990, Science.

[16]  Bruce J. West.  Fractal Forms in Physiology , 1990 .

[17]  K. Horsfield,et al.  Diameters, generations, and orders of branches in the bronchial tree. , 1990, Journal of applied physiology.

[18]  J. Vannimenus,et al.  Combinatorial tools for the analysis of ramified patterns , 1989 .

[19]  P. Meakin.  The Growth of Fractal Aggregates , 1987 .

[20]  A. Goldberger,et al.  Beyond the principle of similitude: renormalization in the bronchial tree. , 1986, Journal of applied physiology.

[21]  Similarity principles for the biology of pelagic animals. , 1983, Proceedings of the National Academy of Sciences of the United States of America.

[22]  T F Sherman,et al.  On connecting large vessels to small. The meaning of Murray's law , 1981, The Journal of general physiology.

[23]  K. Horsfield.  Are diameter, length and branching ratios meaningful in the lung? , 1980, Journal of theoretical biology.

[24]  H. Honda,et al.  BRANCH GEOMETRY AND EFFECTIVE LEAF AREA: A STUDY OF TERMINAUA‐BRANCHING PATTERN. 1. THEORETICAL TREES , 1979 .

[25]  E. Merrill.  Comparison of Mature Leaf Architecture of Three Types in Sorbus L. (Rosaceae) , 1978, Botanical Gazette.

[26]  Eiji Tokunaga,et al.  Consideration on the composition of drainage networks and their evolution , 1978 .

[27]  G Cumming,et al.  Quantitative morphometry of the branching structure of trees. , 1973, Journal of theoretical biology.

[28]  T. McMahon,et al.  Size and Shape in Biology , 1973, Science.

[29]  D E Olson,et al.  Models of the human bronchial tree. , 1971, Journal of applied physiology.

[30]  L. B. Leopold,et al.  Trees and streams: the efficiency of branching patterns. , 1971, Journal of theoretical biology.

[31]  A. N. Strahler.  Quantitative analysis of watershed geomorphology , 1957 .

[32]  R. Horton.  EROSIONAL DEVELOPMENT OF STREAMS AND THEIR DRAINAGE BASINS; HYDROPHYSICAL APPROACH TO QUANTITATIVE MORPHOLOGY , 1945 .

[33]  C D Murray,et al.  The Physiological Principle of Minimum Work: I. The Vascular System and the Cost of Blood Volume. , 1926, Proceedings of the National Academy of Sciences of the United States of America.

[34]  M. Riley,et al.  IN FRACTAL PHYSIOLOGY , 2022 .