A simple mathematical model of allometric exponential growth describes the early three-dimensional growth dynamics of secondary xylem in Arabidopsis roots

Unravelling the specific growth dynamics of key tissues and organs is fundamental to understand how multicellular organisms orchestrate their different growth programmes. In plants, the secondary growth (thickening) of stems and roots provides the mechanical support that plants need to achieve their developmental potential. We used conventional anatomical and microscopy techniques, image-processing software, and quantitative analysis to understand and mathematically describe the growth dynamics of the early developmental stages of secondary xylem (the main tissue developed during secondary growth). Results show that such early developmental stages are characterized by exponential expansion of secondary xylem in three dimensions in the form of an inverted cone, with a power law that describes the relationship between the area of the base and the longitudinal progression (height) of the growing secondary xylem cone over time with a scaling exponent of 2/5: the signature of allometric growth. Our work constitutes a starting point for future modelling of secondary xylem in particular and secondary growth in general.