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Water transport in plants obeys Murray's law


The optimal water transport system in plants should maximize hydraulic conductance (which is proportional to photosynthesis1,2,3,4,5) for a given investment in transport tissue. To investigate how this optimum may be achieved, we have performed computer simulations of the hydraulic conductance of a branched transport system. Here we show that the optimum network is not achieved by the commonly assumed pipe model of plant form6,7,8, or its antecedent, da Vinci's rule9,10. In these representations, the number and area of xylem conduits is constant at every branch rank. Instead, the optimum network has a minimum number of wide conduits at the base that feed an increasing number of narrower conduits distally. This follows from the application of Murray's law, which predicts the optimal taper of blood vessels in the cardiovascular system11. Our measurements of plant xylem indicate that these conduits conform to the Murray's law optimum as long as they do not function additionally as supports for the plant body.

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Figure 1: Transport networks.
Figure 2: Murray's law.
Figure 3: ANOVA P values comparing the sum of the conduit radii raised to the x power (Σrx) between petiolule versus petiole ranks of compound leaves.


  1. Hubbard, R. M., Stiller, V., Ryan, M. G. & Sperry, J. S. Stomatal conductance and photosynthesis vary linearly with plant hydraulic conductance in ponderosa pine. Plant Cell Environ. 24, 113–121 (2001)

    Article  Google Scholar 

  2. Saliendra, N. Z., Sperry, J. S. & Comstock, J. P. Influence of leaf water status on stomatal response to humidity, hydraulic conductance, and soil drought in Betula occidentalis. Planta 196, 357–366 (1995)

    CAS  Article  Google Scholar 

  3. Meinzer, F. C. et al. Environmental and physiological regulation of transpiration in tropical forest gap species: The influence of boundary layer and hydraulic properties. Oecologia 101, 514–522 (1995)

    ADS  CAS  Article  Google Scholar 

  4. Sperry, J. S., Alder, N. N. & Eastlack, S. E. The effect of reduced hydraulic conductance on stomatal conductance and xylem cavitation. J. Exp. Bot. 44, 1075–1082 (1993)

    Article  Google Scholar 

  5. Meinzer, F. C. & Grantz, D. A. Stomatal and hydraulic conductance in growing sugarcane: Stomatal adjustment to water transport capacity. Plant Cell Environ. 13, 383–388 (1990)

    Article  Google Scholar 

  6. West, G. B., Brown, J. H. & Enquist, B. J. A general model for the origin of allometric scaling laws in biology. Science 276, 122–126 (1997)

    CAS  Article  Google Scholar 

  7. West, G. B., Brown, J. H. & Enquist, B. J. A general model for the structure and allometry of plant vascular systems. Nature 400, 664–667 (1999)

    ADS  CAS  Article  Google Scholar 

  8. Enquist, B. J., West, G. B. & Brown, J. H. in Scaling in Biology (eds Brown, J. H. & West, G. B.) 167–198 (Oxford Univ. Press, Oxford, 2000)

    MATH  Google Scholar 

  9. Richter, J. P. The Notebooks of Leonardo da Vinci (1452-1519), Compiled and Edited from the Original Manuscripts (Dover, New York, 1970)

    Google Scholar 

  10. Horn, H. S. in Scaling in Biology (eds Brown, J. H. & West, G. B.) 199–220 (Oxford Univ. Press, Oxford, 2000)

    Google Scholar 

  11. Murray, C. D. The physiological principle of minimum work. I. The vascular system and the cost of blood volume. Proc. Natl Acad. Sci. USA 12, 207–214 (1926)

    ADS  CAS  Article  Google Scholar 

  12. Horn, H. S. The Adaptive Geometry of Trees (Princeton Univ. Press, Princeton, New Jersey, 1971)

    Google Scholar 

  13. Givnish, T. J. in On the Economy of Plant Form and F (ed. Givnish, T. J.) 3–9 (Cambridge Univ. Press, Cambridge, 1986)

    Google Scholar 

  14. Gould, S. J. & Lewontin, R. C. The spandrels of San Marco and the Panglossian paradigm: a critique of the adaptationist programme. Proc. R. Soc. Lond. B 205, 581–598 (1979)

    ADS  CAS  Article  Google Scholar 

  15. Mark, R. Architecture and evolution. Am. Sci. 84, 383–389 (1996)

    ADS  Google Scholar 

  16. Raven, J. A. The evolution of vascular land plants in relation to supracellular transport processes. Adv. Bot. Res. 5, 153–219 (1987)

    MathSciNet  Article  Google Scholar 

  17. Sherman, T. F. On connecting large vessels to small: The meaning of Murray's law. J. Gen. Physiol. 78, 431–453 (1981)

    CAS  Article  Google Scholar 

  18. LaBarbera, M. Principles of design of fluid transport systems in zoology. Science 249, 992–999 (1990)

    ADS  CAS  Article  Google Scholar 

  19. Vogel, S. Life in Moving Fluids: The Physical Biology of Flow (Princeton Univ. Press, Princeton, New Jersey, 1994)

    Google Scholar 

  20. Canny, M. J. The transpiration stream in the leaf apoplast: Water and solutes. Phil. Trans. R. Soc. Lond. B 341, 87–100 (1993)

    ADS  Article  Google Scholar 

  21. Hacke, U. G., Sperry, J. S., Pockman, W. P., Davis, S. D. & McCulloh, K. A. Trends in wood density and structure are linked to prevention of xylem implosion by negative pressure. Oecologia 126, 457–461 (2001)

    ADS  Article  Google Scholar 

  22. Shinozaki, K., Yoda, K., Hozumi, K. & Kira, T. A quantitative analysis of plant form—the pipe model theory: I. Basic analysis. Jpn. J. Ecol. 14, 97–105 (1964)

    Google Scholar 

  23. Sokal, R. R. & Rohlf, F. J. Biometry: The Principles and Practice of Statistics in Biological Research (Freeman, New York, 1995)

    MATH  Google Scholar 

  24. Zimmermann, M. H. in Xylem Structure and the Ascent of Sap (ed. Timell, T. E.) 15–16, 99 (Springer, Berlin, 1983)

    Book  Google Scholar 

  25. Keller, J. B. & Niordson, F. I. The tallest column. J. Math. Mech. 16, 433–446 (1966)

    MathSciNet  MATH  Google Scholar 

  26. McMahon, T. A. Size and shape in biology. Science 179, 1201–1204 (1973)

    ADS  CAS  Article  Google Scholar 

  27. Grafen, A. & Hails, R. Modern Statistics for the Life Sciences (Oxford Univ. Press, Oxford, 2002)

    Google Scholar 

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We thank A. Collopy and M. McCord for assistance in collecting plants. This work was partly supported by Sigma Xi (K.A.M.) and NSF (J.S.S.).

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Correspondence to Katherine A. McCulloh.

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The authors declare that they have no competing financial interests.

Supplementary information


Supplementary Information: Cecil Murray originally derived Murray’s law for the cardiovascular system of animals. Despite the many differences between plant and animal vasculature, including the thickness of conduit walls and the number of conduits per rank, Murray’s derivation can easily be extended to plant xylem. In the Supplementary Material, we provide a derivation of Murray’s law with respect to plants. (DOC 61 kb)

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McCulloh, K., Sperry, J. & Adler, F. Water transport in plants obeys Murray's law. Nature 421, 939–942 (2003).

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