Tracing the opposing assimilate and nutrient flows in live conifer needles.

The vasculature along conifer needles is fundamentally different from that in angiosperm leaves as it contains a unique transfusion tissue inside the bundle sheath. In this study, we used specific tracers to identify the pathway of photoassimilates from mesophyll to phloem, and the opposing pathway of nutrients from xylem to mesophyll. For symplasmic transport we applied esculin to the tip of attached pine needles and followed its movement down the phloem. For apoplasmic transport we let detached needles take up a membrane-impermeable contrast agent and used micro-X-ray computed tomography to map critical water exchange interfaces and domain borders. Microscopy and segmentation of the X-ray data enabled us to render and quantify the functional 3D structure of the water-filled apoplasm and the complementary symplasmic domain. The transfusion tracheid system formed a sponge-like apoplasmic domain that was blocked at the bundle sheath. Transfusion parenchyma cell chains bridged this domain as tortuous symplasmic pathways with strong local anisotropy which, as evidenced by the accumulation of esculin, pointed to the phloem flanks as the preferred phloem-loading path. Simple estimates supported a pivotal role of the bundle sheath, showing that a bidirectional movement of nutrient ions and assimilates is feasible and emphasizing the role of the bundle sheath in nutrient and assimilate exchange.

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.

The glymphatic system: Current understanding and modeling.

We review theoretical and numerical models of the glymphatic system, which circulates cerebrospinal fluid and interstitial fluid around the brain, facilitating solute transport. Models enable hypothesis development and predictions of transport, with clinical applications including drug delivery, stroke, cardiac arrest, and neurodegenerative disorders like Alzheimer's disease. We sort existing models into broad categories by anatomical function: Perivascular flow, transport in brain parenchyma, interfaces to perivascular spaces, efflux routes, and links to neuronal activity. Needs and opportunities for future work are highlighted wherever possible; new models, expanded models, and novel experiments to inform models could all have tremendous value for advancing the field.

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.

The mechanism of sugar export from long conifer needles.

The green leaves of plants are optimised for carbon fixation and the production of sugars, which are used as central units of carbon and energy throughout the plant. However, there are physical limits to this optimisation that remain insufficiently understood. Here, quantitative anatomical analysis combined with mathematical modelling and sugar transport rate measurements were used to determine how effectively sugars are exported from the needle-shaped leaves of conifers in relation to leaf length. Mathematical modelling indicated that phloem anatomy constrains sugar export in long needles. However, we identified two mechanisms by which this constraint is overcome, even in needles longer than 20 cm: (1) the grouping of transport conduits, and (2) a shift in the diurnal rhythm of sugar metabolism and export in needle tips. The efficiency of sugar transport in the phloem can have a significant influence on leaf function. The constraints on sugar export described here for conifer needles are likely to also be relevant in other groups of plants, such as grasses and angiosperm trees.

Sugar export limits size of conifer needles.

Plant leaf size varies by more than three orders of magnitude, from a few millimeters to over one meter. Conifer leaves, however, are relatively short and the majority of needles are no longer than 6 cm. The reason for the strong confinement of the trait-space is unknown. We show that sugars produced near the tip of long needles cannot be exported efficiently, because the pressure required to drive vascular flow would exceed the greatest available pressure (the osmotic pressure). This basic constraint leads to the formation of an inactive region of stagnant fluid near the needle tip, which does not contribute to sugar flow. Remarkably, we find that the size of the active part does not scale with needle length. We predict a single maximum needle size of 5 cm, in accord with data from 519 conifer species. This could help rationalize the recent observation that conifers have significantly smaller leaves than angiosperms, and provide a biophysical explanation for this intriguing difference between the two largest groups of plants.

Double-slit experiment with single wave-driven particles and its relation to quantum mechanics.

In a thought-provoking paper, Couder and Fort [Phys. Rev. Lett. 97, 154101 (2006)] describe a version of the famous double-slit experiment performed with droplets bouncing on a vertically vibrated fluid surface. In the experiment, an interference pattern in the single-particle statistics is found even though it is possible to determine unambiguously which slit the walking droplet passes. Here we argue, however, that the single-particle statistics in such an experiment will be fundamentally different from the single-particle statistics of quantum mechanics. Quantum mechanical interference takes place between different classical paths with precise amplitude and phase relations. In the double-slit experiment with walking droplets, these relations are lost since one of the paths is singled out by the droplet. To support our conclusions, we have carried out our own double-slit experiment, and our results, in particular the long and variable slit passage times of the droplets, cast strong doubt on the feasibility of the interference claimed by Couder and Fort. To understand theoretically the limitations of wave-driven particle systems as analogs to quantum mechanics, we introduce a Schrödinger equation with a source term originating from a localized particle that generates a wave while being simultaneously guided by it. We show that the ensuing particle-wave dynamics can capture some characteristics of quantum mechanics such as orbital quantization. However, the particle-wave dynamics can not reproduce quantum mechanics in general, and we show that the single-particle statistics for our model in a double-slit experiment with an additional splitter plate differs qualitatively from that of quantum mechanics.

Slower phloem transport in gymnosperm trees can be attributed to higher sieve element resistance.

In trees, carbohydrates produced in photosynthesizing leaves are transported to roots and other sink organs over distances of up to 100 m inside a specialized transport tissue, the phloem. Angiosperm and gymnosperm trees have a fundamentally different phloem anatomy with respect to cell size, shape and connectivity. Whether these differences have an effect on the physiology of carbohydrate transport, however, is not clear. A meta-analysis of the experimental data on phloem transport speed in trees yielded average speeds of 56 cm h(-1) for angiosperm trees and 22 cm h(-1) for gymnosperm trees. Similar values resulted from theoretical modeling using a simple transport resistance model. Analysis of the model parameters clearly identified sieve element (SE) anatomy as the main factor for the significantly slower carbohydrate transport speed inside the phloem in gymnosperm compared with angiosperm trees. In order to investigate the influence of SE anatomy on the hydraulic resistance, anatomical data on SEs and sieve pores were collected by transmission electron microscopy analysis and from the literature for 18 tree species. Calculations showed that the hydraulic resistance is significantly higher in the gymnosperm than in angiosperm trees. The higher resistance is only partially offset by the considerably longer SEs of gymnosperms.

Diffusion and bulk flow in phloem loading: a theoretical analysis of the polymer trap mechanism for sugar transport in plants.

Plants create sugar in the mesophyll cells of their leaves by photosynthesis. This sugar, mostly sucrose, has to be loaded via the bundle sheath into the phloem vascular system (the sieve elements), where it is distributed to growing parts of the plant. We analyze the feasibility of a particular loading mechanism, active symplasmic loading, also called the polymer trap mechanism, where sucrose is transformed into heavier sugars, such as raffinose and stachyose, in the intermediary-type companion cells bordering the sieve elements in the minor veins of the phloem. Keeping the heavier sugars from diffusing back requires that the plasmodesmata connecting the bundle sheath with the intermediary cell act as extremely precise filters, which are able to distinguish between molecules that differ by less than 20% in size. In our modeling, we take into account the coupled water and sugar movement across the relevant interfaces, without explicitly considering the chemical reactions transforming the sucrose into the heavier sugars. Based on the available data for plasmodesmata geometry, sugar concentrations, and flux rates, we conclude that this mechanism can in principle function, but that it requires pores of molecular sizes. Comparing with the somewhat uncertain experimental values for sugar export rates, we expect the pores to be only 5%-10% larger than the hydraulic radius of the sucrose molecules. We find that the water flow through the plasmodesmata, which has not been quantified before, contributes only 10%-20% to the sucrose flux into the intermediary cells, while the main part is transported by diffusion. On the other hand, the subsequent sugar translocation into the sieve elements would very likely be carried predominantly by bulk water flow through the plasmodesmata. Thus, in contrast to apoplasmic loaders, all the necessary water for phloem translocation would be supplied in this way with no need for additional water uptake across the plasma membranes of the phloem.

Efficiency of osmotic pipe flows.

We present experiments and theory for flows of sugar or salt solutions in cylindrical tubes with semipermeable walls (hollow fiber membranes) immersed in water, quantifying the strength of the osmotic driving force in relation to the dimensionless parameters that specify the system. The pumping efficiency of these flows is limited by the presence of "unstirred" concentration boundary layers near the tube walls, and our primary aim is to understand and quantify these layers and their effect on the flow. We measure the outlet flow rate Q(out) while varying the inlet flow rate Q(*), concentration c(*), and tube length L, and map out the dependence of the flow rate gain γ=Q(out)/Q(*)-1 on these parameters. A theoretical analysis based on (1) the known velocity field for slow flow in cylindrical porous tubes and (2) a parabolic concentration profile allows us to compute analytically how the flow gain depends on the relative magnitude of radial diffusion and advection as well as the ratio of the osmotic velocity to pumping velocity, in very good agreement with experiments and with no adjustable parameters. Our analysis provides criteria that are useful for optimizing osmotic flow processes in, e.g., water purification devices.

Unifying model of shoot gravitropism reveals proprioception as a central feature of posture control in plants.

Gravitropism, the slow reorientation of plant growth in response to gravity, is a key determinant of the form and posture of land plants. Shoot gravitropism is triggered when statocysts sense the local angle of the growing organ relative to the gravitational field. Lateral transport of the hormone auxin to the lower side is then enhanced, resulting in differential gene expression and cell elongation causing the organ to bend. However, little is known about the dynamics, regulation, and diversity of the entire bending and straightening process. Here, we modeled the bending and straightening of a rod-like organ and compared it with the gravitropism kinematics of different organs from 11 angiosperms. We show that gravitropic straightening shares common traits across species, organs, and orders of magnitude. The minimal dynamic model accounting for these traits is not the widely cited gravisensing law but one that also takes into account the sensing of local curvature, what we describe here as a graviproprioceptive law. In our model, the entire dynamics of the bending/straightening response is described by a single dimensionless "bending number" B that reflects the ratio between graviceptive and proprioceptive sensitivities. The parameter B defines both the final shape of the organ at equilibrium and the timing of curving and straightening. B can be estimated from simple experiments, and the model can then explain most of the diversity observed in experiments. Proprioceptive sensing is thus as important as gravisensing in gravitropic control, and the B ratio can be measured as phenotype in genetic studies.

Modeling the hydrodynamics of Phloem sieve plates.

Sieve plates have an enormous impact on the efficiency of the phloem vascular system of plants, responsible for the distribution of photosynthetic products. These thin plates, which separate neighboring phloem cells, are perforated by a large number of tiny sieve pores and are believed to play a crucial role in protecting the phloem sap from intruding animals by blocking flow when the phloem cell is damaged. The resistance to the flow of viscous sap in the phloem vascular system is strongly affected by the presence of the sieve plates, but the hydrodynamics of the flow through them remains poorly understood. We propose a theoretical model for quantifying the effect of sieve plates on the phloem in the plant, thus unifying and improving previous work in the field. Numerical simulations of the flow in real and idealized phloem channels verify our model, and anatomical data from 19 plant species are investigated. We find that the sieve plate resistance is correlated to the cell lumen resistance, and that the sieve plate and the lumen contribute almost equally to the total hydraulic resistance of the phloem translocation pathway.

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.

Model for polygonal hydraulic jumps.

We propose a phenomenological model for the polygonal hydraulic jumps discovered by Ellegaard and co-workers [Nature (London) 392, 767 (1998); Nonlinearity 12, 1 (1999); Physica B 228, 1 (1996)], based on the known flow structure for the type-II hydraulic jumps with a "roller" (separation eddy) near the free surface in the jump region. The model consists of mass conservation and radial force balance between hydrostatic pressure and viscous stresses on the roller surface. In addition, we consider the azimuthal force balance, primarily between pressure and viscosity, but also including nonhydrostatic pressure contributions from surface tension in light of recent observations by Bush and co-workers [J. Fluid Mech. 558, 33 (2006); Phys. Fluids 16, S4 (2004)]. The model can be analyzed by linearization around the circular state, resulting in a parameter relationship for nearly circular polygonal states. A truncated but fully nonlinear version of the model can be solved analytically. This simpler model gives rise to polygonal shapes that are very similar to those observed in experiments, even though surface tension is neglected, and the condition for the existence of a polygon with N corners depends only on a single dimensionless number φ. Finally, we include time-dependent terms in the model and study linear stability of the circular state. Instability occurs for sufficiently small Bond number and the most unstable wavelength is expected to be roughly proportional to the width of the roller as in the Rayleigh-Plateau instability.

Universality of phloem transport in seed plants.

Since Münch in the 1920s proposed that sugar transport in the phloem vascular system is driven by osmotic pressure gradients, his hypothesis has been strongly supported by evidence from herbaceous angiosperms. Experimental constraints made it difficult to test this proposal in large trees, where the distance between source and sink might prove incompatible with the hypothesis. Recently, the theoretical optimization of the Münch mechanism was shown to lead to surprisingly simple predictions for the dimensions of the phloem sieve elements in relation to that of fast growing angiosperms. These results can be obtained in a very transparent way using a simple coupled resistor model. To test the universality of the Münch mechanism, we compiled anatomical data for 32 angiosperm and 38 gymnosperm trees with heights spanning 0.1-50 m. The species studied showed a remarkable correlation with the scaling predictions. The compiled data allowed calculating stem sieve element conductivity and predicting phloem sap flow velocity. The central finding of this work is that all vascular plants seem to have evolved efficient osmotic pumping units, despite their huge disparity in size and morphology. This contribution extends the physical understanding of phloem transport, and will facilitate detailed comparison between theory and field experiments.

Mechanisms and feasibility of prey capture in ambush-feeding zooplankton.

Many marine zooplankters, particularly among copepods, are "ambush feeders" that passively wait for their prey and capture them by fast surprise attacks. This strategy must be very demanding in terms of muscle power and sensing capabilities, but the detailed mechanisms of the attacks are unknown. Using high-speed video we describe how copepods perform spectacular attacks by precision maneuvering during a rapid jump. We show that the flow created by the attacking copepod is so small that the prey is not pushed away, and that the attacks are feasible because of their high velocity (approximately 100 mm x s(-1)) and short duration (few ms), which leaves the prey no time for escape. Simulations and analytical estimates show that the viscous boundary layer that develops around the attacking copepod is thin at the time of prey capture and that the flow around the prey is small and remains potential flow. Although ambush feeding is highly successful as a feeding strategy in the plankton, we argue that power requirements for acceleration and the hydrodynamic constraints restrict the strategy to larger (> 0.25 mm), muscular forms with well-developed prey perception capabilities. The smallest of the examined species is close to this size limit and, in contrast to the larger species, uses its largest possible jump velocity for such attacks. The special requirements to ambush feeders with such attacks may explain why this strategy has evolved to perfection only a few times among planktonic suspension feeders (few copepod families and chaetognaths).

Osmotically driven flows in microchannels separated by a semipermeable membrane.

We have fabricated lab-on-a-chip systems with microchannels separated by integrated membranes allowing for osmotically driven microflows. We have investigated these flows experimentally by studying the dynamics and structure of the front of a sugar solution travelling in 200 microm wide and 50-200 microm deep microchannels. We find that the sugar front travels at a constant speed, and that this speed is proportional to the concentration of the sugar solution and inversely proportional to the depth of the channel. We propose a theoretical model, which, in the limit of low axial flow resistance, predicts that the sugar front should indeed travel with a constant velocity. The model also predicts an inverse relationship between the depth of the channel and the speed, and a linear relation between the sugar concentration and the speed. We thus find good qualitative agreement between the experimental results and the predictions of the model. Our motivation for studying osmotically driven microflows is that they are believed to be responsible for the translocation of sugar in plants through the phloem sieve element cells. Also, we suggest that osmotic elements can act as on-chip integrated pumps with no movable parts in lab-on-a-chip systems.

Bubble pinch-off in a rotating flow.

We create air bubbles at the tip of a "bathtub vortex" which reaches to a finite depth. The bathtub vortex is formed by letting water drain through a small hole at the bottom of a rotating cylindrical container. The tip of the needlelike surface dip is unstable at high rotation rates and releases bubbles which are carried down by the flow. Using high-speed imaging we find that the minimal neck radius of the unstable tip decreases in time as a power law with an exponent close to 1/3. This exponent was found by Gordillo et al. [Phys. Rev. Lett. 95, 194501 (2005)10.1103/PhysRevLett.95.194501] to govern gas flow driven pinch-off, and indeed we find that the volume oscillations of the tip creates a considerable air flow through the neck. We argue that the Bernoulli pressure reduction caused by this air flow can become sufficient to overcome the centrifugal forces and cause the final pinch-off.

Amplitude equation and long-range interactions in underwater sand ripples in one dimension.

We present an amplitude equation for sand ripples under oscillatory flow in a situation where the sand is moving in a narrow channel and the height profile is practically one dimensional. The equation has the form ht = - epsilon(h-h)+((hx)2-1)hxx-hxxxx+delta((hx)2)xx which, due to the first term, is neither completely local (it has long-range coupling through the average height h) nor has local sand conservation. We argue that this is reasonable and show that the equation compares well with experimental observations in narrow channels. We focus in particular on the so-called doubling transition, a secondary instability caused by the sudden decrease in the amplitude of the water motion, leading to the appearance of a new ripple in each trough. This transition is well reproduced for sufficiently large delta (asymmetry between trough and crest). We finally present surprising experimental results showing that long-range coupling is indeed seen in the initial details of the doubling transition, where in fact two small ripples are initially formed, followed by global symmetry breaking removing one of them.

Polygons on a rotating fluid surface.

We report a novel and spectacular instability of a fluid surface in a rotating system. In a flow driven by rotating the bottom plate of a partially filled, stationary cylindrical container, the shape of the free surface can spontaneously break the axial symmetry and assume the form of a polygon rotating rigidly with a speed different from that of the plate. With water, we have observed polygons with up to 6 corners. It has been known for many years that such flows are prone to symmetry breaking, but apparently the polygonal surface shapes have never been observed. The creation of rotating internal waves in a similar setup was observed for much lower rotation rates, where the free surface remains essentially flat [J. M. Lopez, J. Fluid Mech. 502, 99 (2004). We speculate that the instability is caused by the strong azimuthal shear due to the stationary walls and that it is triggered by minute wobbling of the rotating plate.