Tailor-made composite functions as tools in model choice: the case of sigmoidal vs bi-linear growth profiles

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. 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.

On 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.

Fits 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.