Diversity of funnel plasmodesmata in angiosperms: the impact of geometry on plasmodesmal resistance.

In most plant tissues, threads of cytoplasm, or plasmodesmata, connect the protoplasts via pores in the cell walls. This enables symplasmic transport, for instance in phloem loading, transport and unloading. Importantly, the geometry of the wall pore limits the size of the particles that may be transported, and also (co-)defines plasmodesmal resistance to diffusion and convective flow. However, quantitative information on transport through plasmodesmata in non-cylindrical cell wall pores is scarce. We have found conical, funnel-shaped cell wall pores in the phloem-unloading zone in growing root tips of five eudicot and two monocot species, specifically between protophloem sieve elements and phloem pole pericycle cells. 3D reconstructions by electron tomography suggested that funnel plasmodesmata possess a desmotubule but lack tethers to fix it in a central position. Model calculations showed that both diffusive and hydraulic resistance decrease drastically in conical and trumpet-shaped cell wall pores compared with cylindrical channels, even at very small opening angles. Notably, the effect on hydraulic resistance was relatively larger. We conclude that funnel plasmodesmata generally are present in specific cell-cell interfaces in angiosperm roots, where they appear to facilitate symplasmic phloem unloading. Interestingly, cytosolic sleeves of most plasmodesmata reported in the literature do not resemble annuli of constant diameter but possess variously shaped widenings. Our evaluations suggest that widenings too small for unambiguous identification on electron micrographs may drastically reduce the hydraulic and diffusional resistance of these pores. Consequently, theoretical models assuming cylindrical symmetries will underestimate plasmodesmal conductivities.

Neural networks and robotic microneedles enable autonomous extraction of plant metabolites.

Plant metabolites comprise a wide range of extremely important chemicals. In many cases, like savory spices, they combine distinctive functional properties-deterrence against herbivory-with an unmistakable flavor. Others have remarkable therapeutic qualities, for instance, the malaria drug artemisinin, or mechanical properties, for example natural rubber. We present a breakthrough in plant metabolite extraction technology. Using a neural network, we teach a computer how to recognize metabolite-rich cells of the herbal plant rosemary (Rosmarinus officinalis) and automatically extract the chemicals using a microrobot while leaving the rest of the plant undisturbed. Our approach obviates the need for chemical and mechanical separation and enables the extraction of plant metabolites that currently lack proper methods for efficient biomass use. Computer code required to train the neural network, identify regions of interest, and control the micromanipulator is available as part of the Supplementary Material.

Bending and Stretching of Soft Pores Enable Passive Control of Fluid Flows.

Programmable valves and actuators are widely used in man-made systems to provide sophisticated control of fluid flows. In nature, however, this process is frequently achieved using passive soft materials. Here we study how elastic deformations of cylindrical pores embedded in a flexible membrane enable passive flow control. We develop biomimetic valves with variable pore radius, membrane radius, and thickness. Our experiments reveal a mechanism where small deformations bend the membrane and constrict the pore-thus reducing flow-while larger deformations stretch the membrane, expand the pore, and enhance flow. We develop a theory capturing this highly nonmonotonic behavior, and validate the scaling across a broad range of material and geometric parameters. Our results suggest that intercompartmental flow control in living systems can be encoded entirely in the physical attributes of soft materials. Moreover, this design could enable autonomous flow control in man-made systems.

Convergent evolution of vascular optimization in kelp (Laminariales).

Terrestrial plants and mammals, although separated by a great evolutionary distance, have each arrived at a highly conserved body plan in which universal allometric scaling relationships govern the anatomy of vascular networks and key functional metabolic traits. The universality of allometric scaling suggests that these phyla have each evolved an 'optimal' transport strategy that has been overwhelmingly adopted by extant species. To truly evaluate the dominance and universality of vascular optimization, however, it is critical to examine other, lesser-known, vascularized phyla. The brown algae (Phaeophyceae) are one such group--as distantly related to plants as mammals, they have convergently evolved a plant-like body plan and a specialized phloem-like transport network. To evaluate possible scaling and optimization in the kelp vascular system, we developed a model of optimized transport anatomy and tested it with measurements of the giant kelp, Macrocystis pyrifera, which is among the largest and most successful of macroalgae. We also evaluated three classical allometric relationships pertaining to plant vascular tissues with a diverse sampling of kelp species. Macrocystis pyrifera displays strong scaling relationships between all tested vascular parameters and agrees with our model; other species within the Laminariales display weak or inconsistent vascular allometries. The lack of universal scaling in the kelps and the presence of optimized transport anatomy in M. pyrifera raises important questions about the evolution of optimization and the possible competitive advantage conferred by optimized vascular systems to multicellular phyla.

Scaling of phloem structure and optimality of photoassimilate transport in conifer needles.

The phloem vascular system facilitates transport of energy-rich sugar and signalling molecules in plants, thus permitting long-range communication within the organism and growth of non-photosynthesizing organs such as roots and fruits. The flow is driven by osmotic pressure, generated by differences in sugar concentration between distal parts of the plant. The phloem is an intricate distribution system, and many questions about its regulation and structural diversity remain unanswered. Here, we investigate the phloem structure in the simplest possible geometry: a linear leaf, found, for example, in the needles of conifer trees. We measure the phloem structure in four tree species representing a diverse set of habitats and needle sizes, from 1 (Picea omorika) to 35 cm (Pinus palustris). We show that the phloem shares common traits across these four species and find that the size of its conductive elements obeys a power law. We present a minimal model that accounts for these common traits and takes into account the transport strategy and natural constraints. This minimal model predicts a power law phloem distribution consistent with transport energy minimization, suggesting that energetics are more important than translocation speed at the leaf level.

Pico gauges for minimally invasive intracellular hydrostatic pressure measurements.

Intracellular pressure has a multitude of functions in cells surrounded by a cell wall or similar matrix in all kingdoms of life. The functions include cell growth, nastic movements, and penetration of tissue by parasites. The precise measurement of intracellular pressure in the majority of cells, however, remains difficult or impossible due to their small size and/or sensitivity to manipulation. Here, we report on a method that allows precise measurements in basically any cell type over all ranges of pressure. It is based on the compression of nanoliter and picoliter volumes of oil entrapped in the tip of microcapillaries, which we call pico gauges. The production of pico gauges can be accomplished with standard laboratory equipment, and measurements are comparably easy to conduct. Example pressure measurements are performed on cells that are difficult or impossible to measure with other methods.

Measurement of flow velocity and inference of liquid viscosity in a microfluidic channel by fluorescence photobleaching.

We present a simple, noninvasive method for simultaneous measurement of flow velocity and inference of liquid viscosity in a microfluidic channel. We track the dynamics of a sharp front of photobleached fluorescent dye using a confocal microscope and measure the intensity at a single point downstream of the initial front position. We fit an exact solution of the advection diffusion equation to the fluorescence intensity recovery curve to determine the average flow velocity and the diffusion coefficient of the tracer dye. The dye diffusivity is correlated to solute concentration to infer rheological properties of the liquid. This technique provides a simple method for simultaneous elucidation of flow velocity and liquid viscosity in microchannels.

Competition between slicing and buckling underlies the erratic nature of paper cuts.

By enabling the dissemination and storage of information, paper has been central to human culture for more than a millennium. Its use is, however, associated with a common injury: the paper cut. Surprisingly, the physics underpinning a flexible sheet of paper slicing into soft tissues remains unresolved. In particular, the unpredictable occurrence of paper cuts, often restricted to a limited thickness range, has not been explained. Here we visualize and quantify the motion, deformation, and stresses during paper cuts, uncovering a remarkably complex relationship between cutting, geometry, and material properties. A model based on the hypothesis that a competition between slicing and buckling controls the probability of initiating a paper cut is developed and successfully validated. This explains why paper with a specific thickness is most hazardous (65µm, corresponding, e.g., to dot matrix paper) and suggests a probabilistic interpretation of irregular occurrence of paper cuts. Stimulated by these findings, we finally show how a recyclable cutting tool can harness the surprising power of paper.

The fluidic memristor as a collective phenomenon in elastohydrodynamic networks.

Fluid flow networks are ubiquitous and can be found in a broad range of contexts, from human-made systems such as water supply networks to living systems like animal and plant vasculature. In many cases, the elements forming these networks exhibit a highly non-linear pressure-flow relationship. Although we understand how these elements work individually, their collective behavior remains poorly understood. In this work, we combine experiments, theory, and numerical simulations to understand the main mechanisms underlying the collective behavior of soft flow networks with elements that exhibit negative differential resistance. Strikingly, our theoretical analysis and experiments reveal that a minimal network of nonlinear resistors, which we have termed a 'fluidic memristor', displays history-dependent resistance. This new class of element can be understood as a collection of hysteresis loops that allows this fluidic system to store information, and it can be directly used as a tunable resistor in fluidic setups. Our results provide insights that can inform other applications of fluid flow networks in soft materials science, biomedical settings, and soft robotics, and may also motivate new understanding of the flow networks involved in animal and plant physiology.

Physical limits to leaf size in tall trees.

Leaf sizes in angiosperm trees vary by more than 3 orders of magnitude, from a few mm to over 1 m. This large morphological freedom is, however, only expressed in small trees, and the observed leaf size range declines with tree height, forming well-defined upper and lower boundaries. The vascular system of tall trees that distributes the products of photosynthesis connects distal parts of the plant and forms one of the largest known continuous microfluidic distribution networks. In biological systems, intrinsic properties of vascular systems are known to constrain the morphological freedom of the organism. We show that the limits to leaf size can be understood by physical constraints imposed by intrinsic properties of the carbohydrate transport network. The lower boundary is set by a minimum energy flux, and the upper boundary is set by a diminishing gain in transport efficiency.

Elastohydrodynamic autoregulation in soft overlapping channels.

Controlling fluid flow from an unsteady source is a challenging problem that is relevant in both living and man-made systems. Animals have evolved various autoregulatory mechanisms to maintain homeostasis in vital organs. This keeps the influx of nutrients essentially constant and independent of the perfusion pressure. Up to this point, the autoregulation processes have primarily been ascribed to active mechanisms that regulate vessel size, thereby adjusting the hydraulic conductance in response to, e.g., sensing of wall shear stress. We propose an alternative elastohydrodynamic mechanism based on contacting soft vessels. Inspired by Starling's resistor, we combine experiments and theory to study the flow of a viscous liquid through a self-intersecting soft conduit. In the overlapping region, the pressure difference between the two channel segments can cause one pipe segment to dilate while the other is compressed. If the tissue is sufficiently soft, this mode of fluid-structure interactions can lead to flow autoregulation. Our experimental observations compare well to a predictive model based on low-Reynolds-number fluid flow and linear elasticity. Implications for conduit arrangement and passive autoregulation in organs and limbs are discussed.

Locally optimal geometry for surface-enhanced diffusion.

Molecular diffusion in bulk liquids proceeds according to Fick's law, which stipulates that the particle current is proportional to the conductive area. This constrains the efficiency of filtration systems in which both selectivity and permeability are valued. Previous studies have demonstrated that interactions between the diffusing species and solid boundaries can enhance or reduce particle transport relative to bulk conditions. However, only cases that preserve the monotonic relationship between particle current and conductive area are known. In this paper, we expose a system in which the diffusive current increases when the conductive area diminishes. These examples are based on the century-old theory of a charged particle interacting with an electrical double layer. This surprising discovery could improve the efficiency of filtration and may advance our understanding of biological pore structures.

Pavement cells distinguish touch from letting go.

A micro-cantilever technique applied to individual leaf epidermis cells of intact Arabidopsis thaliana and Nicotiana tabacum synthesizing genetically encoded calcium indicators (R-GECO1 and GCaMP3) revealed that compressive forces induced local calcium peaks that preceded delayed, slowly moving calcium waves. Releasing the force evoked significantly faster calcium waves. Slow waves were also triggered by increased turgor and fast waves by turgor drops in pressure probe tests. The distinct characteristics of the wave types suggest different underlying mechanisms and an ability of plants to distinguish touch from letting go.

Astrocyte endfeet may theoretically act as valves to convert pressure oscillations to glymphatic flow.

The glymphatic system of cerebrospinal fluid transport through the perivascular spaces of the brain has been implicated in metabolic waste clearance, neurodegenerative diseases and in acute neurological disorders such as stroke and cardiac arrest. In other biological low-pressure fluid pathways such as in veins and the peripheral lymphatic system, valves play an important role in ensuring the flow direction. Though fluid pressure is low in the glymphatic system and directed bulk flow has been measured in pial and penetrating perivascular spaces, no valves have yet been identified. Valves, which asymmetrically favour forward flow to backward flow, would imply that the considerable oscillations in blood and ventricle volumes seen in magnetic resonance imaging could cause directed bulk flow. Here, we propose that astrocyte endfeet may act as such valves using a simple elastic mechanism. We combine a recent fluid mechanical model of viscous flow between elastic plates with recent measurements of in vivo elasticity of the brain to predict order of magnitude flow-characteristics of the valve. The modelled endfeet are effective at allowing forward while preventing backward flow.

Analytic solutions and universal properties of sugar loading models in Münch phloem flow.

The transport of sugars in the phloem vascular system of plants is believed to be driven by osmotic pressure differences according to the Münch hypothesis. Thus, the translocation process is viewed as a passive reaction to the active sugar loading in the leaves and sugar unloading in roots and other places of growth or storage. The modelling of the loading and unloading mechanism is thus a key ingredient in the mathematical description of such flows, but the influence of particular choices of loading functions on the translocation characteristics is not well understood. Most of the work has relied on numerical solutions, which makes it difficult to draw general conclusions. Here, we present analytic solutions to the Münch-Horwitz flow equations when the loading and unloading rates are assumed to be linear functions of the concentration, thus allowing them to depend on the local osmotic pressure. We are able to solve the equations analytically for very small and very large Münch numbers (e.g., very small and very large viscosity) for the flow velocity and sugar concentration as a function of the geometric and material parameters of the system. We further show, somewhat surprisingly, that the constant loading case can be solved along the same lines and we speculate on possible universal properties of different loading and unloading functions applied in the literature.

Sugar loading is not required for phloem sap flow in maize plants.

Phloem transport of photoassimilates from leaves to non-photosynthetic organs, such as the root and shoot apices and reproductive organs, is crucial to plant growth and yield. For nearly 90 years, evidence has been generally consistent with the theory of a pressure-flow mechanism of phloem transport. Central to this hypothesis is the loading of osmolytes, principally sugars, into the phloem to generate the osmotic pressure that propels bulk flow. Here we used genetic and light manipulations to test whether sugar import into the phloem is required as the driving force for phloem sap flow. Using carbon-11 radiotracer, we show that a maize sucrose transporter1 (sut1) loss-of-function mutant has severely reduced export of carbon from photosynthetic leaves (only ~4% of the wild type level). Yet, the mutant remarkably maintains phloem pressure at ~100% and sap flow speeds at ~50-75% of those of wild type. Potassium (K+) abundance in the phloem was elevated in sut1 mutant leaves. Fluid dynamic modelling supports the conclusion that increased K+ loading compensated for decreased sucrose loading to maintain phloem pressure, and thereby maintained phloem transport via the pressure-flow mechanism. Furthermore, these results suggest that sap flow and transport of other phloem-mobile nutrients and signalling molecules could be regulated independently of sugar loading into the phloem, potentially influencing carbon-nutrient homoeostasis and the distribution of signalling molecules in plants encountering different environmental conditions.

Viscous energy dissipation in slender channels with porous or semipermeable walls.

We study the viscous dissipation in pipe flows in long channels with porous or semipermeable walls, taking into account both the dissipation in the bulk of the channel and in the pores. We give simple closed-form expressions for the dissipation in terms of the axially varying flow rate Q(x) and the pressure p(x), generalizing the well-known expression W[over ̇]=QΔp=RQ^{2} for the case of impenetrable walls with constant Q, pressure difference Δp between the ends of the pipe and resistance R. When the pressure p_{0} outside the pipe is constant, the result is the straightforward generalization W[over ̇]=Δ[(p-p_{0})Q]. Finally, applications to osmotic flows are considered.

Concentration-Responsive Soft Valve for Osmotic Flow Control.

We propose a novel osmotic soft valve consisting of an osmosis membrane and hydrogel films. In our osmotic valve system, material selectivity is determined by the osmosis membrane, and the hydrogel film, which deforms depending on the ion concentration of the surrounding solution, controls the passage area of the membrane. Independently controlling the material selectivity and permeability allowed us to design an osmotic soft valve with an osmotic flow rate that increases with osmotic pressure at low pressures but decreases with osmotic pressure at high pressures. We demonstrate a representative application of our hydrogel valve system in a portable power generator utilizing reverse electrodialysis (RED). As the permeability varied with concentration, the hydrogel valve was able to maintain the electric power of the RED for 30 min with only an ∼10% change. Our study provides techniques to build osmotic soft valves that can serve as gating membranes in various osmosis and dialysis systems.

Plasmodesmata and the problems with size: Interpreting the confusion.

Plant tissues exhibit a symplasmic organization; the individual protoplasts are connected to their neighbors via cytoplasmic bridges that extend through pores in the cell walls. These bridges may have diameters of a micrometer or more, as in the sieve pores of the phloem, but in most cell types they are smaller. Historically, botanists referred to cytoplasmic bridges of all sizes as plasmodesmata. The meaning of the term began to shift when the transmission electron microscope (TEM) became the preferred tool for studying these structures. Today, a plasmodesma is widely understood to be a 'nano-scale' pore. Unfortunately, our understanding of these nanoscopic channels suffers from methodological limitations. This is exemplified by the fact that state-of-the-art EM techniques appear to reveal plasmodesmal pore structures that are much smaller than the tracer molecules known to diffuse through these pores. In general, transport processes in pores that have dimensions in the size range of the transported molecules are governed by different physical parameters than transport process in the macroscopic realm. This can lead to unexpected effects, as experience in nanofluidic technologies demonstrates. Our discussion of problems of size in plasmodesma research leads us to conclude that the field will benefit from technomimetic reasoning - the utilization of concepts developed in applied nanofluidics for the interpretation of biological systems.

Directionality of Plasmodesmata-Mediated Transport in Arabidopsis Leaves Supports Auxin Channeling.

The plant hormone auxin serves as central regulator of growth and development. Auxin transporters in the plasma membrane are assumed to define tissue-level patterns of auxin distribution [1, 2]. However, auxin is small enough to diffuse through the plasmodesmata that connect neighboring cells [3], presenting an alternative pathway, whose contribution to auxin transport remained largely unexplored [4]. Here, photoactivation microscopy [5, 6] was used to measure the capacity for small-molecule diffusion in the epidermis of Arabidopsis thaliana leaves. In the elongated epidermis cells covering the midrib and petiole, the plasmodesmata-mediated cell-wall permeability was found to be several times higher in the longitudinal than in the transverse direction. The physiological relevance of this asymmetry was tested through quantification of the shade-avoidance response, which depends on auxin transport from the leaf tip to the petiole in the abaxial side of the leaf [7], with the hypothesis that directionality of diffusion supplements transporter-mediated auxin movement [8]. Triggering the response by auxin application at the tip led to stronger leaf movement in wild-type plants than in gsl8 mutants [9], which lack the callose synthase necessary to establish directionality. The results match the predictions of a mathematical model of auxin transport based on the permeabilities measured in wild-type and mutant plants. It is concluded that plasmodesmata permeability can be selectively modulated within a plant cell and that the conferred directionality in diffusion can influence the tissue-specific distribution patterns of small molecules, like auxin. VIDEO ABSTRACT.