The decline of hydraulic conductivity with suction in plants

In the vein of understanding more on plants hydraulics I started from the (McDowell et al., 2008)} paper which is as much informative as imprecise in its two or three formulas which do not much with the concepts conveyedby the figures and the text. So definitely it is an inspiring paper but I would not start from it. Instead I would probably use as a first reading {(Venturas et al., 2017)} that is more recent and more consistent and which I would couple with Lehnebach et al., 2018), to read as acomplement. In this post, however, let's concentrate on the conductivity of stems decline with stems suction. In literature, it was found todecline with stem pressure according to a sigmoid form (I would say, an exceeding probability like s function).In literature various form were used to parametrize this behavior. Notably:

• the exponential-sigmoid (Pammenter & der Willigen., 1998)

$$ K(\psi) = K_{max} \left( 1 - \frac{1}{1+ \exp( a(\psi-b) )}\right) $$

• the Weibull (Rawlings \& Cure, 1985), Neufeld et al., 1992)

$$ K(\psi) = K_{max} e^{ - \left(\frac{\psi}{\alpha} \right)^\beta} $$

• the Gomperz (Mencuccini \& Comstock, 1997)

$$ K(\psi) = 1 - a e^{-b e^{-c \psi}} $$

• polynomial function (Pockman et al., 1995)
• the power law (Wang et al., 2017)

$$ K(\psi) = \left(\frac{1}{2}\right)^{\left(\frac{\psi}{\psi_{50}}\right)^b}$$
However, for historical reasons, this decline is modeled~in plant physiology literature using PLC curves which represent the Percent Loss Conductivity. PLC have a one to one relation with hydraulic conductivity
which is given by the formula:
$$ K(\psi) = K_{max}\left(1-\frac{PLC(\psi)}{100}\right) $$
Some researchers found useful to treat differently the parameterization as, for instance, in Ogle et al., 2009, an indication that was taken in Duursma, 2017 for implementing their R software fitplc. The latter paper is interesting also for having tried to evaluate the error of estimation and the parameters they reproduce in their Table 1 for some plant species.

References

• Nate McDowell, William T. Pockman, Craig D. Allen, David D. Breshears, Neil Cobb, Thomas Kolb, Jennifer Plaut, John Sperry, Adam West, David G. Williams, Enrico A. Yepez. Mechanisms of plant survival and mortality during drought: why do some plants survive while others succumb to drought?. New Phytologist178, 719–739 Wiley, 2008. Link
• Martin D. Venturas, John S. Sperry, Uwe G. Hacke. Plant xylem hydraulics: What we understand current research, and future challenges. Journal of Integrative Plant Biology 59, 356–389 Wiley, 2017. Link
• Romain Lehnebach, Robert Beyer, Véronique Letort, Patrick Heuret. The pipe model theory half a century on: a review. Annals of Botany 121, 773–795 Oxford University Press (OUP), 2018. Link
• N. W. Pammenter, C. Van der Willigen.. A Mathematical and Statistical Analysis of the Curves Illustrating Vulnerability of Xylem to Cavitation. Tree Physiology 18 (1998).
• J. O. Rawlings, W. W. Cure. The Weibull Function as a Dose-Response Model to Describe Ozone Effects on Crop Yields 1. Crop Science 25, 807–814 Wiley, 1985. Link
• Howard S. Neufeld, David A. Grantz, Frederick C. Meinzer, Guillermo Goldstein, Gayle M. Crisosto, Carlos Crisosto. Genotypic Variability in Vulnerability of Leaf Xylem to Cavitation in Water-Stressed and Well-Irrigated Sugarcane. Plant Physiology 100, 1020–1028 American Society of Plant Biologists (ASPB), 1992.Link
• M. Mencuccini, J. Comstock. Vulnerability to cavitation in populations of two desert species,Hymenoclea salsolaandAmbrosia dumosa from different climatic regions. Journal of Experimental Botany 48, 1323–1334 Oxford University Press (OUP), 1997. Link
• William T. Pockman, John S. Sperry, James W. OLeary. Sustained and significant negative water pressure in xylem. Nature 378, 715–716 Springer Science and Business Media LLC, 1995. Link
• Han Wang, I. Colin Prentice, Trevor F. Keenan, Tyler W. Davis, Ian J. Wright, William K. Cornwell, Bradley J. Evans, Changhui Peng. Towards a universal model for carbon dioxide uptake by plants. Nature Plants 3, 734–741 Springer Science and Business Media LLC, 2017. Link
• Kiona Ogle, Jarrett J. Barber, Cynthia Willson, Brenda Thompson. Hierarchical statistical modeling of xylem vulnerability to cavitation. New Phytologist 182, 541–554 Wiley, 2009. Link
• R. & Choat Duursma. fitplc - an R package to fit hydraulic vulnerability curves. J. Plant Hydraul. (2017)