A lognormal distribution of the lengths of terminal twigs on self-similar branches of elm trees

Scaling the leaf length-times-width equation to predict total leaf area of shoots

BACKGROUND AND AIMS

An individual plant consists of different-sized shoots, each of which consists of different-sized leaves. To predict plant-level physiological responses from the responses of individual leaves, modelling this within-shoot leaf size variation is necessary. Within-plant leaf trait variation has been well investigated in canopy photosynthesis models but less so in plant allometry. Therefore, integration of these two different approaches is needed.

METHODS

We focused on an established leaf-level relationship that the area of an individual leaf lamina is proportional to the product of its length and width. The geometric interpretation of this equation is that different-sized leaf laminas from a single species share the same basic form. Based on this shared basic form, we synthesized a new length-times-width equation predicting total shoot leaf area from the collective dimensions of leaves that comprise a shoot. Furthermore, we showed that several previously established empirical relationships, including the allometric relationships between total shoot leaf area, maximum individual leaf length within the shoot and total leaf number of the shoot, can be unified under the same geometric argument. We tested the model predictions using five species, all of which have simple leaves, selected from diverse taxa (Magnoliids, monocots and eudicots) and from different growth forms (trees, erect herbs and rosette herbs).

KEY RESULTS

For all five species, the length-times-width equation explained within-species variation of total leaf area of a shoot with high accuracy (R2 > 0.994). These strong relationships existed despite leaf dimensions scaling very differently between species. We also found good support for all derived predictions from the model (R2 > 0.85).

CONCLUSIONS

Our model can be incorporated to improve previous models of allometry that do not consider within-shoot size variation of individual leaves, providing a cross-scale linkage between individual leaf-size variation and shoot-size variation.

Evaluation of the Normality Assumption in Meta-Analyses

Random-effects meta-analysis is one of the mainstream methods for research synthesis. The heterogeneity in meta-analyses is usually assumed to follow a normal distribution. This is actually a strong assumption, but one that often receives little attention and is used without justification. Although methods for assessing the normality assumption are readily available, they cannot be used directly because the included studies have different within-study standard errors. Here we present a standardization framework for evaluation of the normality assumption and examine its performance in random-effects meta-analyses with simulation studies and real examples. We use both a formal statistical test and a quantile-quantile plot for visualization. Simulation studies show that our normality test has well-controlled type I error rates and reasonable power. We also illustrate the real-world significance of examining the normality assumption with examples. Investigating the normality assumption can provide valuable information for further analysis or clinical application. We recommend routine examination of the normality assumption with the proposed framework in future meta-analyses.

The arrangement of lateral veins along the midvein of leaves is not related to leaf phyllotaxis

Positions of leaves along a stem usually adhere to a genetically determined, species-specific pattern known as a leaf phyllotaxis. We investigated whether the arrangement of lateral secondary veins along primary midveins adhered to a species-specific pattern that resembled an alternate or opposite phyllotaxis. We analyzed the venation of temperate dicotyledonous species from different taxonomic groups and chose 18 woody and 12 herbaceous species that have reticulated leaf venation. The arrangement of the lateral veins was neither alternate nor opposite for any of the species. Lateral vein arrangements were instead mixtures of symmetric and asymmetric patterns. Our results show that lateral vein arrangements are related neither to stem-level leaf phyllotaxis (alternate vs. opposite) nor to life form (woody vs. herbaceous). Our results are therefore generally consistent with the canalization hypothesis that the locations of lateral veins are not completely specified genetically prior to leaf formation.

A lognormal distribution of the lengths of terminal twigs on self-similar branches of elm trees

Lognormal distributions and self-similarity are characteristics associated with a wide range of biological systems. The sequential breakage model has established a link between lognormal distributions and self-similarity and has been used to explain species abundance distributions. To date, however, there has been no similar evidence in studies of multicellular organismal forms. We tested the hypotheses that the distribution of the lengths of terminal stems of Japanese elm trees (Ulmus davidiana), the end products of a self-similar branching process, approaches a lognormal distribution. We measured the length of the stem segments of three elm branches and obtained the following results: (i) each occurrence of branching caused variations or errors in the lengths of the child stems relative to their parent stems; (ii) the branches showed statistical self-similarity; the observed error distributions were similar at all scales within each branch and (iii) the multiplicative effect of these errors generated variations of the lengths of terminal twigs that were well approximated by a lognormal distribution, although some statistically significant deviations from strict lognormality were observed for one branch. Our results provide the first empirical evidence that statistical self-similarity of an organismal form generates a lognormal distribution of organ sizes.