# A Nonlinear Fitting Method Provides Strong Support for Geometric Series of Stomatal Area in 12 Magnoliaceae Species

## TL;DR

A new nonlinear fitting method improves the accuracy of modeling stomatal area sequences in 12 Magnoliaceae species, supporting the geometric series hypothesis.

## Contribution

A novel nonlinear regression method (Method-2) is introduced for fitting stomatal area sequences, significantly reducing prediction errors compared to prior methods.

## Key findings

- Method-2 achieved 99.6% of MAPE values ≤5%, compared to 52.4% for the previous method.
- The geometric series hypothesis for stomatal area sequences is strongly supported across 12 Magnoliaceae species.
- Interspecific differences in model parameters and MAPE values were small, indicating consistent behavior across species.

## Abstract

Stomatal pore area and density determine the capacity for gas exchange between the leaf interior and the atmosphere. Stomatal area is given by the profile formed by two guard cells, and the cumulative stomatal area characterizes the area of leaf surface occupied by stomata. The areas of all stomata captured in a micrograph are sorted in ascending order to form a sequence, which is referred to as a sequence of stomatal area here. In total, 360 leaves of 12 Magnoliaceae species with 30 leaves for each species were sampled. For each leaf, two 662 μm × 444 μm fields of view (micrographs) of stomata were captured on the right leaf width axis. In each micrograph, the length and width of each stoma were measured, and the area of the stoma was determined using the product of stomatal length and width multiplied by a proportionality coefficient. Stomatal area sequences of Magnoliaceae in the constant field of view were found to follow a geometric series (GS). Prior studies estimated the common ratio of the GS as the mean of the quotients of any two adjacent terms, and estimated the first term as the mean of the first terms (i.e., the smallest stomatal area) represented by the quotient of each term and the estimated common ratio to a power of the order of the term minus 1, which is referred to as Method-1. However, it produced large prediction errors for some stomatal area sequences. In the present study, the nonlinear regression was used to fit the stomatal area sequences using the common ratio and the first term as two model parameters (Method-2). We compared the two methods using the mean absolute percent error (MAPE, ≤5% considered as a good fit) values of the 720 stomatal micrographs from the 12 Magnoliaceae species. The goodness of fit of Method-2 was better than that of Method-1 (52.4% MAPE values were ≤5% for Method-1 and 99.6% for Method-2). There were significant variations in the estimated common ratios, as well as the estimated first terms and the MAPE values across the 12 Magnoliaceae species, but overall, the interspecific differences in the MAPE values were small. We conclude that the GS hypothesis for the stomatal area sequences of the 12 Magnoliaceae species was further strengthened by the new method. This method further provides a valuable approach for the calculation of total stomatal area per unit leaf area.