Yoël Forterre

Slow, fast and furious: understanding the physics of plant movements

The ability of plants to move is central to many physiological processes from development to tropisms, from nutrition to reproduction. The movement of plants or plant parts occurs over a wide range of sizes and time scales. This review summarizes the main physical mechanisms plants use to achieve motility, highlighting recent work at the frontier of biology and physics on rapid movements. Emphasis is given to presenting in a single framework pioneering biological studies of water transport and growth with more recent physics research on poroelasticity and mechanical instabilities. First, the basic osmotic and hydration/dehydration motors are described that contribute to movement by growth and reversible swelling/shrinking of cells and tissues. The speeds of these water-driven movements are shown to be ultimately limited by the transport of water through the plant body. Some plant structures overcome this hydraulic limit to achieve much faster movement by using a mechanical instability. The principle is to impose an energy barrier to the system, which can originate from geometrical constraint or matter cohesion, allowing elastic potential energy to be stored until the barrier is overcome, then rapidly transformed into kinetic energy. Three of these rapid motion mechanisms have been elucidated recently and are described here: the snapping traps of two carnivorous plants, the Venus flytrap and Utricularia, and the catapult of fern sporangia. Finally, movement mechanisms are reconsidered in the context of the timescale of important physiological processes at the cellular and molecular level.

Flow of dense granular material: towards simple constitutive laws

Recent experiments and numerical simulations of dry granular flows suggest that a simple rheological description in terms of a friction coefficient varying both with shear rate and pressure through a dimensionless inertial number may be sufficient to capture the major properties of granular flows. In this paper we first present the empirical constitutive laws and their interpretation using dimensional analysis, before analysing the prediction for different flow configurations. The successes and limits of the approach are discussed based on comparison with recent studies.

Generating Helices in Nature

A general model accounts for different types of helical shapes that can result from bilayers of stretchable, bendable sheets. Macroscopic helical structures formed by organisms include seashells, horns, plant tendrils, and seed pods (see the figure, panel A). The helices that form are chiral; like wood screws, they have a handedness. Some are helicoids, twisted helices with saddle-like curvature and a straight centerline; others are cylindrical helices with cylindrical curvature and a helical centerline. Studies of the mechanisms underlying the formation of helicoid or helical ribbons and of the transitions between these structures (1–4) have left an important question unanswered: How do the molecular organization of the material and its global geometrical features interact to create a diversity of helical shapes? On page 1726 of this issue, Armon et al. (5) explore the rich phenomenology associated with slender strips made of mutually opposing “molecular” layers, taking a singular botanical structure—the Bauhinia seed pod—as their inspiration. They show that a single component, namely a flat strip with a saddle-like intrinsic curvature, is sufficient to generate a wide variety of helical shapes.

Drop impact of yield-stress fluids

The normal impact of a drop of yield-stress fluid on a flat rigid surface is investigated experimentally. Using di! erent model fluids (polymer microgels, clay suspensions) and impacted surfaces (partially wettable, super-hydrophobic), we find a rich variety of impact regimes from irreversible viscoplastic coating to giant elastic spreading and recoil. A minimal model of inertial spreading, taking into account an elastoviscoplastic rheology, allows explaining in a single framework the di! erent regimes and scaling laws. In addition, semi-quantitative predictions for the spread factor are obtained when the measured rheological parameters of the fluid (elasticity, yield stress, viscosity) are injected into the model. Our study o! ers a means to probe the short-time rheology of yield-stress fluids and highlights the role of elasticity on the unsteady hydrodynamics of these complex fluids. Movies are available with the online version of the paper (go to journals.cambridge.org/flm).

Revealing the frictional transition in shear-thickening suspensions

Significance The sudden and severe increase in the viscosity of certain suspensions above an onset stress is one of the most spectacular phenomena observed in complex fluids. This shear thickening, which has major implications for industry, is a long-standing puzzle in soft-matter physics. Recently, a frictional transition was conjectured to cause this phenomenon. Using experimental concepts from granular physics, we provide direct evidence that such suspensions are frictionless under low confining pressure, which is key to understanding their shear-thickening behavior. Shear thickening in dense particulate suspensions was recently proposed to be driven by the activation of friction above an onset stress needed to overcome repulsive forces between particles. Testing this scenario represents a major challenge because classical rheological approaches do not provide access to the frictional properties of suspensions. Here we adopt a different strategy inspired by pressure-imposed configurations in granular flows that specifically gives access to this information. By investigating the quasi-static avalanche angle, compaction, and dilatancy effects in different nonbuoyant suspensions flowing under gravity, we demonstrate that particles in shear-thickening suspensions are frictionless under low confining pressure. Moreover, we show that tuning the range of the repulsive force below the particle roughness suppresses the frictionless state and also the shear-thickening behavior of the suspension. These results, which link microscopic contact physics to the suspension macroscopic rheology, provide direct evidence that the recent frictional transition scenario applies in real suspensions.

Unifying Impacts in Granular Matter from Quicksand to Cornstarch

A sharp transition between liquefaction and transient solidification is observed during impact on a granular suspension depending on the initial packing fraction. We demonstrate, via high-speed pressure measurements and a two-phase modeling, that this transition is controlled by a coupling between the granular pile dilatancy and the interstitial fluid pressure generated by the impact. Our results provide a generic mechanism for explaining the wide variety of impact responses in particulate media, from dry quicksand in powders to impact hardening in shear-thickening suspensions like cornstarch.

Biomechanics of rapid movements in plants: poroelastic measurements at the cell scale

From a biomechanical perspective, plants offer a fascinat- ing example of living systems capable of producing non- muscular movements (Skotheim and Mahadevan 2005; Dumais and Forterre, to be published). Although most of these movements are slow, some compete in speed with those observed in the animal kingdom and are involved in essential functions such as seed/pollen dispersal, defence and nutrition. Of these spectacular examples that have long fascinated scientists, the Venus flytrap (Figure 1(a)), for which the leaves snap together in a fraction of second to capture insects, has long been a paradigm for study. Recently, we have shown that this motion involves a snap- buckling instability due to the shell-like geometry of the leaves of the trap (Forterre et al. 2005). However, the origin of the active movement used by the plant to cross the instability threshold remains unknown. More generally, the physical mechanisms involved in rapid plant movements remain poorly understood, especially at the cell and tissue scale. Two main assumptions are found in the literature: (i) a rapid flow between the cells due to changes in osmotic pressure (Hill and Findlay 1981), (ii) a rapid cell expansion due to mechanical modifications (softening) in the cell wall (Williams and Bennett 1982). In both cases, the high-water pressure inside the plant cells is believed to play a central role. Our aim in this study was to measure in real time in-vivo mechanical and hydraulic properties of Venus flytrap cells (cell wall elasticity, cell membrane per- meability and cell water pressure) and to investigate the possible mechanisms for movements at the cell scale. To this end, we use a microfluidic pressure probe device, which applies a water flux inside a single cell while measuring the cell water pressure simultaneously (Steudle 1993).

A new scenario for gravity detection in plants: the position sensor hypothesis

The detection of gravity plays a fundamental role during the growth and evolution of plants. Although progress has been made in our understanding of the molecular, cellular and physical mechanisms involved in the gravity detection, a coherent scenario consistent with all the observations is still lacking. In this special issue article, we discuss recent experiments showing that the response to inclination of shoots is independent of the gravity intensity, meaning that the gravity sensor detects an inclination and not a force. This result questions some of the commonly accepted hypotheses and leads to propose a new position sensor hypothesis. The implications of this new scenario are discussed in light of the different observations available in the literature.

Universal poroelastic mechanism for hydraulic signals in biomimetic and natural branches

Significance Plants are sessile organisms without nerves. As such, they have developed specific mechanisms to carry information rapidly throughout their body in response to mechanical stimuli. Recently, it has been suggested that the first stage of this long-distance signaling could be the propagation of hydraulic signals induced by the mechanical deformation of the plant tissue (bending), but the physical origin of this hydromechanical coupling remains a conundrum. Here, we address this issue by combining experiments on natural tree branches and soft biomimetic beams with modeling. We reveal a generic nonlinear mechanism responsible for the generation of hydraulic pulses induced by bending in poroelastic branches. Our study gives a physical basis for long-distance communication in plants based on fast hydraulic signals. Plants constantly undergo external mechanical loads such as wind or touch and respond to these stimuli by acclimating their growth processes. A fascinating feature of this mechanical-induced growth response is that it can occur rapidly and at long distance from the initial site of stimulation, suggesting the existence of a fast signal that propagates across the whole plant. The nature and origin of the signal is still not understood, but it has been recently suggested that it could be purely mechanical and originate from the coupling between the local deformation of the tissues (bending) and the water pressure in the plant vascular system. Here, we address the physical origin of this hydromechanical coupling using a biomimetic strategy. We designed soft artificial branches perforated with longitudinal liquid-filled channels that mimic the basic features of natural stems and branches. In response to bending, a strong overpressure is generated in the channels that varies quadratically with the bending curvature. A model based on a mechanism analogous to the ovalization of hollow tubes enables us to predict quantitatively this nonlinear poroelastic response and identify the key physical parameters that control the generation of the pressure pulse. Further experiments conducted on natural tree branches reveal the same phenomenology. Once rescaled by the model prediction, both the biomimetic and natural branches fall on the same master curve, enlightening the universality of our poroelastic mechanism for the generation of hydraulic signals in plants.

Granular Media: The granular liquid

Most granular flows encountered in nature and industry lie between the quasistatic and gaseous regimes seen in the previous chapters. In this intermediate ‘liquid’ regime, particles remain closely packed and interact both by collision and through long-lived contacts. Understanding and modelling the flow of dense granular media is challenging and many questions remain to be answered, despite important advances having been made during the last decade. In this chapter, we first present the basic features of dense granular flows (Section 6.1), before focusing on the rheology of this peculiar liquid (Section 6.2). A phenomenological constitutive law that is based on dimensional analysis is presented, in which the medium is described as a viscoplastic fluid with a frictional behaviour. The success and limitations of this approach are then discussed, in particular close to the solid–liquid transition where complex collective behaviours are observed. The second part of the chapter presents a hydrodynamic description of dense flows that is valid for a shallow layer flowing under gravity (the Saint-Venant equations) (Section 6.3). This depth-averaged approach enables one to gather the complex rheology into a single basal friction term and is commonly used in geophysics to describe rock avalanches and landslides. We close the chapter with a presentation of the phenomenon of size segregation, which occurs when the medium is composed of particles of different sizes. The consequences of segregation for polydisperse granular flows in various configurations are presented (Section 6.4).