Abstract

BackgroundRoots are the classical model system to study the organization and dynamics of organ growth zones. Profiles of the velocity of root elements relative to the apex have generally been considered to be sigmoidal. However, recent high-resolution measurements have yielded bi-linear profiles, suggesting that sigmoidal profiles may be artifacts caused by insufficient spatio-temporal resolution. The decision whether an empirical velocity profile follows a sigmoidal or bi-linear distribution has consequences for the interpretation of the underlying biological processes. However, distinguishing between sigmoidal and bi-linear curves is notoriously problematic. A mathematical function that can describe both types of curve equally well would allow them to be distinguished by automated curve-fitting.ResultsOn the basis of the mathematical requirements defined, we created a composite function and tested it by fitting it to sigmoidal and bi-linear models with different noise levels (Monte-Carlo datasets) and to three experimental datasets from roots of Gypsophila elegans, Aurinia saxatilis, and Arabidopsis thaliana. Fits of the function proved robust with respect to noise and yielded statistically sound results if care was taken to identify reasonable initial coefficient values to start the automated fitting procedure. Descriptions of experimental datasets were significantly better than those provided by the Richards function, the most flexible of the classical growth equations, even in cases in which the data followed a smooth sigmoidal distribution.ConclusionFits of the composite function introduced here provide an independent criterion for distinguishing sigmoidal and bi-linear growth profiles, but without forcing a dichotomous decision, as intermediate solutions are possible. Our function thus facilitates an unbiased, multiple-working hypothesis approach. While our discussion focusses on kinematic growth analysis, this and similar tailor-made functions will be useful tools wherever models of steadily or abruptly changing dependencies between empirical parameters are to be compared.

Highlights

  • Roots are the classical model system to study the organization and dynamics of organ growth zones

  • In contrast to the sigmoidal models, which dominate text-books and previous research reports, bi-linear growth profiles imply that cells switch between two distinct expansion modes when going through their period of elongation (Fig. 1)

  • Biological meaning enters our analyses via the geometry of the fitted curve, which reflects the characteristics of the time-courses of physiological parameters experienced by cells traversing the growth zone (Figs. 1, 3)

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Summary

Introduction

Roots are the classical model system to study the organization and dynamics of organ growth zones. Profiles of the velocity of root elements relative to the apex have generally been considered to be sigmoidal. The decision whether an empirical velocity profile follows a sigmoidal or bilinear distribution has consequences for the interpretation of the underlying biological processes. Distinguishing between sigmoidal and bi-linear curves is notoriously problematic. Kinematic growth analysis aims at the quantitative description of spatial growth patterns to provide a basis for the study of developmental mechanisms [1,2]. As the term kinematic indicates, this approach focuses on the movement of parts of a growing organ relative to each other. The concepts for kinematic growth analysis were laid out half-a-century ago for the root, which was (page number not for citation purposes)

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