Maximum-likelihood estimation of xylem vessel length distributions.

Vessel length is an important functional trait for plant hydraulics, because it determines the ratio of flow resistances posed by lumen and pit membranes and hence controls xylem hydraulic efficiency. The most commonly applied methods to estimate vessel lengths are based on the injection of silicon or paint into cut-off stem segments. The number of stained vessels in a series of cross-sections in increasing distance from the injection point is then counted. The resulting infusion profiles are used to estimate the vessel length distribution using one of several statistical algorithms. However, the basis of these algorithms has not been systematically analysed using probability theory. We derive a general mathematical expression for the expected shape of the infusion profile for a given vessel length distribution, provide analytic solutions for five candidate distributions (exponential, Erlang(2), gamma, Weibull, and log-normal), and present maximum likelihood estimators for the parameters of these distributions including implementations in R based on two potential sampling schemes (counting all injected vessels or counting the injected and empty vessels in a random subset of each cross-section). We then explore the performance of these estimators relative to other methods with Monte Carlo experiments. Our analysis demonstrates that most published methods estimate the conditional length distribution of vessels that cross an injection point, which is a size-biased version of the overall length distribution in the stem. We show the mathematical relationship between these distributions and provide methods to estimate either of them. According to our simulation experiments, vessel length distribution was best described by the more flexible models, especially the Weibull distribution. In simulations, the estimators were able to recover the parameters of the vessel length distribution if its functional form was known, achieving an overlap of 90% or more between the true and predicted length distribution when counting no more than 500 injected vessels in 10 cross-sections. This sample size nowadays can easily be reached with the help of automated image analysis.