A unified morphoelastic rod model with application to growth-induced coiling, waving, and skewing of plant roots

Abstract

The shapes of roots exhibit distinctive patterns, which attracted significant attempts at an explanation. We develop a geometrically exact, elastic, three-dimensional morphoelastic rod model of root elongation coupled with control mechanisms representing circumnutation, gravitropism, and thigmotropism. Possible forms of the evolution equations for the intrinsic curvature of the rod, representing these control mechanisms are discussed. By specifying the evolution rules of intrinsic curvatures, the model boils down to a sequential solution of two-point boundary value problems in order to compute the evolving root shape. We simulate root growth on hard soil surfaces with different angles of inclination. Systematic computations show that the morphology is predominantly determined by the model parameters associated with the control mechanisms and by the angle of inclination. Predictions of the new model recover experimentally observed root coiling, waving, and skewing of Arabidopsis thaliana. Our results suggest that the primary mechanism behind root coiling, waving, and skewing is the interplay between circumnutation and gravitropism as slope angle is varied. Thigmotropism has a moderate effect on the emerging shapes.