Fitting functions to leaf hydraulic or stomatal responses using maximum likelihood model selection



Christine Scoffoni, Lawren Sack

Author affiliations/Institutions

University of California, Los Angeles


The purpose of this protocol is to fit different functions to leaf hydraulic vulnerability curves, and select the best fit model (with the lowest AICc score).


Leaf hydraulic conductance and stomatal conductance decline with dehydration, and studies have shown that the shape of this decline varies across species, with some being linear, and other non-linear (Brodribb & Holbrook, 2006, Guyot, Scoffoni & Sack, 2012, Scoffoni, McKown, Rawls & Sack, 2012). Studies constructing leaf hydraulic vulnerability curves have typically fitted functions that appeared appropriate subjectively (See Table S1 in Scoffoni et al., 2012). This protocol is based on maximum likelihood model selection (MLMS). MLMS estimates model parameters that maximize the log-likelihood of observing the parameter values given the data, and further, allows selection of the model from a set of candidate models using the Akaike Information Criterion (AIC), i.e., choosing the model with the optimal balance between maximizing the log-likelihood of the fitted parameters and maximizing parsimony in the number of parameters (Burnham & Anderson, 2002). This approach enables standardization of curve fitting. We select among five different functions that have previously been used in the literature—linear, exponential, logistic, sigmoidal and Weibull (See Scoffoni et al., 2012).


Units, terms, definitions

Ψleaf = leaf water potential (MPa)

Kleaf = leaf hydraulic conductance (mmol m-2 s-1 MPa-1)

gs = stomatal conductance (mmol m-2 s-1)

PLC = percent loss of conductivity

P50 = leaf water potential at 50% loss of leaf conductivity (MPa)

P80 = leaf water potential at 80% loss of leaf conductivity (MPa)

Kmax (or gs,max) = maximum leaf hydraulic (or stomatal) conductance (i.e., for fully hydrated leaf)


1) First you need to create a .csv document with in the first column your Ψleaf values, and in the second column you Kleaf (or gs) values. The first column should have the following headers: psi, Kleaf (or gs)

2) Call the .csv document “PLC data” (or you can change the name of it, but be sure to change it in the script as well).

3) Run the scripts for the different functions.

4) For each function, enter in the Excel document the values for the different parameters, r2, slope, and AIC score and standard errors for the different parameters.

5) The best fit model will be the one with the lowest AIC score.

6) If two or more models have AIC scores within two of each other, then select the one with the highest r2 value.

7) Formulas to calculate Kmax, (or gs,max) P50 and P80 are given in the excel spreadsheet for each function, but be sure to use only the values from the best fit model!

Literature references

Brodribb T.J. & Holbrook N.M. (2006) Declining hydraulic efficiency as transpiring leaves desiccate: two types of response. Plant Cell and Environment, 29, 2205-2215.

Burnham K.P. & Anderson D.R. (2002) Model selection and multi-model inference, 2nd ed. Springer, NY.

Guyot G., Scoffoni C. & Sack L. (2012) Combined impacts of irradiance and dehydration on leaf hydraulic conductance: insights into vulnerability and stomatal control. Plant, Cell & Environment, 35, 857-871.

Scoffoni C., McKown A.D., Rawls M. & Sack L. (2012) Dynamics of leaf hydraulic conductance with water status: quantification and analysis of species differences under steady-state. Journal of Experimental Botany, 63, 643-658.