Mathematical models for biological motility: From peristaltic crawling to plant nutations

In this thesis we propose mathematical models for the motility of one-dimensional crawlers moving along a line and for growing slender plant organs, which are applied to the study of peristaltic crawling and nutations of plant shoots, respectively. The first chapter contains a theoretical analysis of metameric worm-like robotic crawlers, and it investigates optimal actuation strategies. Our main result is that peristalsis, i.e., muscle extension and contraction waves propagating along the body, is an optimal actuation strategy for locomotion. We give a rigorous mathematical proof of this result by solving analytically the optimal control problem in the regime of small deformations. We show that phase coordination arises from the geometric symmetry of a 1D system, exactly in the periodic case and approximately, due to edge-effects, in the case of a crawler of finite length. In the second chapter we introduce the general framework of morphoelastic rods to model elongating slender plant organs. This chapter is intended as preparatory to the third one, where we derive a rod model that is exploited to investigate the role of mechanical deformations in circumnutating plant shoots. We show that, in the absence of endogenous cues, spontaneous oscillations might arise as system instabilities when a loading parameter exceeds a critical value. Moreover, when oscillations of endogenous nature are present, their relative importance with respect to the ones associated with the former mechanism varies in time, as the biomechanical properties of the shoot change. Our findings suggest that the relative importance of exogenous versus endogenous oscillations is an emergent property of the system, and that elastic deformations play a crucial role in this kind of phenomena.