Loss of stability: a new look at the physics of cell wall behavior during plant cell growth

In this article we investigate aspects of turgor-driven plant cell growth within the framework of a model derived from the Eulerian concept of instability. In particular we explore the relationship between cell geometry and cell turgor pressure by extending loss of stability theory to encompass cylindrical cells. Beginning with an analysis of the three-dimensional stress and strain of a cylindrical pressure vessel, we demonstrate that loss of stability is the inevitable result of gradually increasing internal pressure in a cylindrical cell. The turgor pressure predictions based on this model differ from the more traditional viscoelastic or creep-based models in that they incorporate both cell geometry and wall mechanical properties in a single term. To confirm our predicted working turgor pressures, we obtained wall dimensions, elastic moduli, and turgor pressures of sequential internodal cells of intact Chara corallina plants by direct measurement. The results show that turgor pressure predictions based on loss of stability theory fall within the expected physiological range of turgor pressures for this plant. We also studied the effect of varying wall Poisson's ratio nu on extension growth in living cells, showing that while increasing elastic modulus has an understandably negative effect on wall expansion, increasing Poisson's ratio would be expected to accelerate wall expansion.     

An analytical model of the hydraulic aspects of stomatal dynamics

An analytical model of the hydraulic aspects of stomatal dynamics is formulated in this paper. The model consists of a coupled system of non-linear, ordinary differential equations, written in terms of water potentials, hydrostatic pressures, osmotic potentials, water vapor resistances and water fluxes. The model is validated by comparisons with the experimental literature. Numerical solutions of the model show qualitative agreement with most known stomatal responses.Stomatal opening in the model is dependent on the interaction of the guard and subsidiary cells in the following manner. Pore opening is initiated by a rise in the guard cell hydrostatic pressure. As the stomate opens, transpiration increases, causing the cell wall water potential to drop. The drop in cell wall water potential then causes the subsidiary cell pressure to drop, opening is accelerated, and the stomate literally “pops” open. Simulated opening proceeds in two distinct phases: a stress phase and a motor phase. During the stress phase, guard cell pressure rises but the pore remains closed. The motor phase commences when the guard cell pressure has risen sufficiently to initiate pore opening, beyond which point opening progresses rapidly.Hydropassive stomatal movements are found to be insufficient to regulate water loss at low leaf water potentials. Stable, hydraulically-based oscillations in stomatal aperture are shown in the model by the existence of a stable limit cycle. The period of these oscillations is strongly influenced by the cell membrane hydraulic conductivity. An increased conductivity results in a shorter period oscillation. Environmental conditions promoting oscillatory behavior are in qualitative agreement with the experimental literature.     

Effects of transmural pressure and wall shear stress on LDL accumulation in the arterial wall: a numerical study using a multilayered model

The accumulation of low-density lipoprotein (LDL) is recognized as one of the main contributors in atherogenesis. Mathematical models have been constructed to simulate mass transport in large arteries and the consequent lipid accumulation in the arterial wall. The objective of this study was to investigate the influences of wall shear stress and transmural pressure on LDL accumulation in the arterial wall by a multilayered, coupled lumen-wall model. The model employs the Navier-Stokes equations and Darcy's Law for fluid dynamics, convection-diffusion-reaction equations for mass balance, and Kedem-Katchalsky equations for interfacial coupling. To determine physiologically realistic model parameters, an optimization approach that searches optimal parameters based on experimental data was developed. Two sets of model parameters corresponding to different transmural pressures were found by the optimization approach using experimental data in the literature. Furthermore, a shear-dependent hydraulic conductivity relation reported previously was adopted. The integrated multilayered model was applied to an axisymmetric stenosis simulating an idealized, mildly stenosed coronary artery. The results show that low wall shear stress leads to focal LDL accumulation by weakening the convective clearance effect of transmural flow, whereas high transmural pressure, associated with hypertension, leads to global elevation of LDL concentration in the arterial wall by facilitating the passage of LDL through wall layers.     

Parenchymal tethering, airway wall stiffness, and the dynamics of bronchoconstriction.

We do not yet have a good quantitative understanding of how the force-velocity properties of airway smooth muscle interact with the opposing loads of parenchymal tethering and airway wall stiffness to produce the dynamics of bronchoconstriction. We therefore developed a two-dimensional computational model of a dynamically narrowing airway embedded in uniformly elastic lung parenchyma and compared the predictions of the model to published measurements of airway resistance made in rats and rabbits during the development of bronchoconstriction following a bolus injection of methacholine. The model accurately reproduced the experimental time-courses of airway resistance as a function of both lung inflation pressure and tidal volume. The model also showed that the stiffness of the airway wall is similar in rats and rabbits, and significantly greater than that of the lung parenchyma. Our results indicate that the main features of the dynamical nature of bronchoconstriction in vivo can be understood in terms of the classic Hill force-velocity relationship operating against elastic loads provided by the surrounding lung parenchyma and an airway wall that is stiffer than the parenchyma.     

High pressure phase transition and elastic properties of covalent heavy rare-earth Antimonides

In the present paper, we report an investigation into the high-pressure structural phase transition of rare earth antimonides (DySb and ErSb). A modified interaction potential model (MIPM) (including the covalency effect) has been developed. Phase transition pressures are associated with a sudden collapse in volume, indicating the occurrence of a first order phase transition. At compressed volumes, these compounds are found in the CsCl phase. The phase transition pressures and associated volume collapses obtained from the potential model developed here show a generally better agreement with available experimental data than others available in the literature. The elastic constants and bulk modulus are also reported. Our results are, in general, in good agreement with experimental and theoretical data where available, and provide predictions where data are unavailable.     

The specific hydraulic conductivity of bovine serum albumin.

Previous studies of extracellular matrix hydraulic conductivity have characterized the flow resistance of glycosaminoglycans, proteoglycans and collagen. This work focuses on serum albumin, present in significant quantities in many connective tissues, but not previously considered for its role in determining connective tissue flow resistance. The specific hydraulic conductivity of bovine serum albumin solutions, as a function of concentration, was calculated from sedimentation and ultrafiltration data available in the literature. A rigid particle hydrodynamic model compared favorably with these results. Experimental measurements on an albumin ultrafiltration cell were in agreement with this model (within experimental error); furthermore, the experimental data confirmed the theoretical prediction that there is no (or negligible) pressure drop through the concentration polarization layer. Use of the hydrodynamic model for albumin specific hydraulic conductivity with literature values for the hindrance of albumin when passing through a glycosaminoglycan (GAG) matrix allows an estimate of the relative importance of the albumin on tissue hydraulic conductivity: in non-cartilaginous tissues with moderate GAG concentrations, tissue levels of albumin can generate flow resistance effects comparable to those of the GAGs, although well less than the flow resistance of these tissues.     

Estimates of the peak pressures in bone pore water.

The maximum pore fluid pressures due to uniaxial compression are determined for both the vascular porosity (Haversian and Volkmann's canals) and the lacunar-canalicular porosity of live cortical bone. It is estimated that the peak pore water pressure will be 19 percent of the applied axial stress in the vascular porosity and 12 percent of the applied axial stress in the lacunar-canalicular porosity for an impulsive step loading. However, the estimated relaxation time for the vascular porosity (1.36 microseconds) is three orders of magnitude faster than that estimated for the lacunar-canalicular porosity (4.9 ms). Thus, under physiological loading, which has a stress rise time generally larger than 1 ms, pressures higher than the vascular pressure cannot be sustained in the vascular porosity due to the swift pressure relaxation in this porosity (unless the fluid drainage through the boundary is obstructed). The model also predicts a slight hydraulic stiffening of the bulk modulus due to longer draining time of the lacunar-canalicular porosity. The undrained bulk modulus is 6 percent higher than the drained bulk modulus in this case.     

A model of non-uniform lung parenchyma distortion

A finite element model of mammalian lung parenchyma is used to study the effect of large non-uniform distortions on lung elastic behaviour. The non-uniform distortion is a uni-axial stretch from an initial state of uniform pressure expansion. For small distortions, the parenchymal properties are linearly isotropic and described by two elastic moduli. However, for large distortions, the parenchyma has anisotropic non-linear elastic properties described by five independent elastic moduli dependent on the degree of distortion; they are computed for a range of distortions and initial pressures. Ez, the Young's modulus in the direction of stretch, increases significantly with distortion (epsilon(z)) while Ex, the Young's modulus in the plane perpendicular to the stretch, is approximately constant. The greater the initial pressure, the bigger the difference between the two moduli at larger distortion strains. The shear modulus G(xz) is approximately independent of degree of distortion except at the highest initial pressure. The Poisson's ratio, nu(xz) is approximately constant with distortion strain for lower initial pressures, but increases significantly with epsilon(z) at higher pressures. Model predictions of the relation between G(xz) and initial uniform inflation pressure show a good correlation with reported experimental data for small distortion strains in a range of species. The model also exhibits similar behaviour to the experimentally measured uni-axial large deformations of a tri-axially pre-loaded block of parenchyma (Hoppin et al., 1975, Journal of Applied Physiology 39, 742-751).     

Theoretical and experimental exclusion of errors in the determination of the elasticity and water transport parameters of plant cells by the pressure probe technique

The volumetric elastic modulus of the cell wall and the hydraulic conductivity of the cell membranes were measured on ligatured compartments of different sizes of Chara corallina internodes using the pressure probe technique. The ratio between intact cell surface area and the area of puncture in the cell wall and membrane introduced by the microcapillary of the pressure probe was varied over a large range by inserting microcapillaries of widely varying diameters in different sized compartments. The relationship of the elastic modulus and the hydraulic conductivity to turgor pressure was independent of the ratio of intact cell surface area to the area of injury. The increase in the hydraulic conductivity below 2 bar turgor pressure and the volume dependence of the elastic modulus were shown to be the same as those observed in intact nonligatured cells. Theoretical considerations of the possible influence of injury of the cell wall and cell membrane around the inserted microcapillary on the measurement of the water transport and cell wall parameters do not explain the experimental findings. Thus, mechanical artifacts, if at all present, are too small to account for the observed dependence of the hydraulic conductivity and the elastic modulus on turgor pressure. The pressure probe technique thus represents an accurate method for measuring water transport parameters in both giant algal cells and in tissue cells of higher plants.     

Biomechanical adaptation of porcine carotid vascular smooth muscle to hypo and hypertension in vitro

Previous research in arterial remodeling in response to changes in blood pressure seldom included both hyper- and hypotension. To compare the effects of low and high pressure on arterial remodeling and vascular smooth muscle tone and performance, we have utilized an in vitro model. Porcine carotid arteries were cultured for 3 days at 30 and 170mmHg and compared to controls cultured at 100mmHg for 1 and 3 days. On the first and last day of culture, pressure-diameter and pressure-wall thickness curves were measured under normal smooth muscle tone using a high-resolution ultrasonic device. Last-day experiments included measurements where vascular smooth muscle was contracted or totally relaxed. From the data wall cross-sectional area, Hudetz elastic modulus and a contraction index related to the diameter reduction under normal smooth muscle tone were calculated. We found that although wall cross-sectional area (indicating wall mass) did not change much, Hudetz elastic modulus was significantly reduced in the 3-day hypotension group. Inspection of the wall contraction index suggests that this is due to a reduction in the vascular smooth muscle tone. Further, the peak of contraction index was found to be shifted to higher pressures in the 3-day 170mmHg group. We conclude that vascular smooth muscle performance adapts to both hypo- and hypertension at short time scales and can alter the biomechanics of the vascular wall in vitro.     

Hydraulic properties of living late metaxylem and interactions between transpiration and xylem pressure in maize

A new approach to study dynamic interactions between transpiration and xylem pressure in intact plants is presented. Pressure probe measurements were preformed in living (immature) late metaxylem of maize roots rather than in adjacent mature xylem. This eliminated technical limitations related to the measurement of negative pressures. Water relations of single cells showed that turgor and volumetric elastic modulus were significantly larger in living metaxylem than in cortical cells; hydraulic conductivity was similar in both types of root cells. Increasing transpiration induced an immediate decrease of xylem pressure, and vice versa. Turgor in the living metaxylem could be continuously recorded for more than 1 h. The relationship between xylem pressure and transpiration yielded a root hydraulic resistance of 1.3 x 10⁹ MPa s m^-3. Control experiments indicated that the response of living xylem in the positive pressure range essentially paralleled that of mature root xylem in the negative range. In mature xylem, pressures as low as -0.55 MPa were recorded for short periods (several minutes). Several tests verified that the pressure probe was in contact with mature xylem during the measurements of tensions. The results demonstrate convincingly that transpiration generates an effective driving force for water uptake in roots, a central feature of the cohesion theory.Key words: Hydraulic conductivity, negative pressure, root development, turgor, water transport, Zea mays.      

Effect of boundary conditions,impact loading and hydraulic stiffening on femoral fracture strength

Patient specific quantitative CT (QCT) imaging data together with the finite element (FE) method may provide an accurate prediction of a patient's femoral strength and fracture risk. Although numerous FE models investigating femoral fracture strength have been published, there is little consent on the effect of boundary conditions, dynamic loading and hydraulic strengthening due to intra-medullary pressure on the predicted fracture strength. We developed a QCT-derived FE model of a proximal femur that included node-specific modulus assigned based on the local bone density. The effect of three commonly used boundary conditions published in literature were investigated by comparing the resulting strain field due to an applied fracture load. The models were also augmented with viscoelastic material properties and subject to a realistic impact load profile to determine the effect of dynamic loads on the strain field. Finally, the effect of hydraulic strengthening was investigated by including node specific permeability and performing a coupled pore diffusion and stress analysis of the FE model. Results showed that all boundary conditions yield the same strain field patterns, but peak strains were 22% lower and fracture load was 18% higher when loaded at the greater trochanter than when loaded at the femoral head. Comparison of the dynamic models showed that material viscoelasticity was important, but inertial effects (vibration and shock) were not. Finally, pore pressure changes did not cause significant hydraulic strengthening of bone under fall impact loading.     

Effect of age on mechanical properties of rat mesenteric small arteries

With aging, large arteries become stiffer and systolic blood pressure consequently increases. Less is known, however, about the age-related change in mechanics of small resistance arteries. The aim of this study was to determine whether aging plays a role in the stiffening of the small mesenteric arteries of rats. Intra-arterial systolic, diastolic, mean and pulse pressures were measured in male Wistar rats aged 2, 4, 15 and 26 months. The passive mechanical properties of the wall of isolated perfused and pressurized arterial segments of mesenteric small arteries were also investigated. Intra-arterial systolic, diastolic and mean blood pressures tended to decrease with age and were significantly lower in the oldest rats (26-month-old group). Pulse pressure was significantly higher in the 15- and 26-month-old groups than in the two younger groups. Under isobaric conditions, increasing age is associated with an outward hypertrophic remodeling of the mesenteric arteries. Under relaxed conditions, incremental distensibility in response to increasing intravascular pressure did not change with aging. As a function of strain (under isometric conditions), stress shifted to the left as age increased, indicating an age-related vascular stiffening. Under isobaric conditions or in relation to wall stress, the elastic modulus was greater in the adult 15-month-old rats than in the younger rats. These findings suggest that distensibility seems to be preserved with aging, despite stiffness of the wall components, probably by arterial wall geometric adaptation, which limits the pulse pressure damage. It is interesting to note that elastic modulus in mesenteric arteries from the oldest rats (26-month-old), examined in relation to wall stress and intravascular pressure, did not differ from that of the youngest rats, thus suggesting that elasticity of wall components had been restored.     

Validation of computational fluid dynamics methodology used for human upper airway flow simulations

An anatomically accurate human upper airway model was constructed from multiple magnetic resonance imaging axial scans. This model was used to conduct detailed Computational Fluid Dynamics (CFD) simulations during expiration, to investigate the fluid flow in the airway regions where obstruction could occur. An identical physical model of the same airway was built using stereo lithography. Pressure and velocity measurements were conducted in the physical model. Both simulations and experiments were performed at a peak expiratory flow rate of 200 L/min. Several different numerical approaches within the FLUENT commercial software framework were used in the simulations; unsteady Large Eddy Simulation (LES), steady Reynolds-Averaged Navier-Stokes (RANS) with two-equation turbulence models (i.e. k?ε, standard k?ω, and k?ω Shear Stress Transport (SST)) and with one-equation Spalart–Allmaras model. The CFD predictions of the average wall static pressures at different locations along the airway wall were favorably compared with the experimental data. Among all the approaches, standard k?ω turbulence model resulted in the best agreement with the static pressure measurements, with an average error of ～20% over all ports. The highest positive pressures were observed in the retroglossal regions below the epiglottis, while the lowest negative pressures were recorded in the retropalatal region. The latter is a result of the airflow acceleration in the narrow retropalatal region. The largest pressure drop was observed at the tip of the soft palate. This location has the smallest cross section of the airway. The good agreement between the computations and the experimental results suggest that CFD simulations can be used to accurately compute aerodynamic flow characteristics of the upper airway.     

A novel micropipet method for measuring the bending modulus of vesicle membranes.

A theoretical model and an experiment are presented for determining the bending modulus of a bilayer vesicle membrane. The vesicle is held with a pipet having a radius between 1 and 2 microns, and the tension in the membrane is changed by changing the suction pressure. Then the vesicle membrane is deformed by aspirating it into a smaller pipet having a radius on the order of 0.5 microns. The relationship between the suction pressures in the two pipets is found to be linear, as predicted by the theoretical model. The curvature of the vesicle membrane at the pipet orifice and the bending modulus are found with the help of the model from the slope and the intercept of the linear experimental relationship between the suction pressures in the two pipets. The bending modulus for the two SOPC membranes studied in these experiments was found to be either 0.6 or 1.15 x 10(-19) J, which is similar to the values measured previously.     

Examination of elastic non-uniformity in the arterial system using a hydraulic model

Pressure wave propagation has been examined in a model artery with spatially varying compliance. Although results were affected by viscous losses, appropriate allowance for such losses produced agreement between experimental findings and predictions of linear wave transmission theory. Particularly, the ability of non-uniformity of the tube wall to generate amplification of the pressure wave was confirmed. However, extrapolation to the physiological situation suggests that reflections from discrete sites in peripheral beds have a greater effect on pressure wave propagation than does elastic non-uniformity of major vessels. A theoretical analysis has demonstrated that the effects of elastic non-uniformity can be interpreted as the integrated effects of infinitesimal reflections from each progressive increment in wall stiffness.     

Modelling the mechanical properties of single suspension-cultured tomato cells

BACKGROUND AND AIMS: The relationship between composition and structure of plant primary cell walls, and cell mechanical properties is not fully understood, partly because intrinsic properties of walls such as Young's modulus cannot be obtained readily. The aim of this work is to show that Young's modulus of walls of single suspension-cultured tomato cells can be determined by modelling force-deformation data. METHODS: The model simulates the compression of a cell between two flat surfaces, with the cell treated as a liquid-filled sphere with thin compressible walls. The cell wall and membrane were taken to be permeable, but the compression was so fast that water loss could be neglected in the simulations. Force-deformation data were obtained by compressing the cells in micromanipulation experiments. RESULTS:Good fits were obtained between the model and low-strain experimental data, using the modulus and initial inflation of the cell as adjustable parameters. The mean Young's modulus for 2-week-old cells was found to be 2.3 +/- 0.2 GPa at pH 5. This corresponds to an instantaneous bulk modulus of elasticity of approx. 7 MPa, similar to a value found by the pressure probe method. However, Young's modulus is a better parameter, as it should depend only on the composition and structure of the cell wall, not on bulk cell behaviour. This new method has been used to show that Young's modulus of cultured tomato cell walls is at its lowest at pH 4.5, the pH optimum for expansin activity. CONCLUSIONS:The linear elastic model is very suitable for estimating wall Young's modulus from micromanipulation experiments on single tomato cells. This is a powerful method for determining cell wall material properties.     

New evidence for large negative xylem pressures and their measurement by the pressure chamber method

Pressure probe measurements have been interpreted as showing that xylem pressures below c. –0.4 MPa do not exist and that pressure chamber measurements of lower negative pressures are invalid. We present new evidence supporting the pressure chamber technique and the existence of xylem pressures well below –0.4 MPa. We deduced xylem pressures in water-stressed stem xylem from the following experiment: (1) loss of hydraulic conductivity in hydrated stem xylem (xylem pressure = atmospheric pressure) was induced by forcing compressed air into intact xylem conduits; (2) loss of hydraulic conductivity from cavitation and embolism in dehydrating stems was measured, and (3) the xylem pressure in dehydrated stems was deduced as being equal and opposite to the air pressure causing the same loss of hydraulic conductivity in hydrated stems. Pressures determined in this way are only valid if cavitation was caused by air entering the xylem conduits (air-seeding). Deduced xylem pressure showed a one-to-one correspondence with pressure chamber measurements for 12 species (woody angiosperms and gymnosperms); data extended to c. –10 MPa. The same correspondence was obtained under field conditions in Betula occidentalis Hook., where pressure differences between air- and water-filled conduits were induced by a combination of in situ xylem water pressure and applied positive air pressure. It is difficult to explain these results if xylem pressures were above –0.4 MPa, if the pressure chamber was inaccurate, and if cavitation occurred by some mechanism other than air-seeding. A probable reason why the pressure probe does not register large negative pressures is that, just as cavitation within the probe limits its calibration to pressures above c. –0.5 MPa, cavitation limits its measurement range in situ.     

Structural changes in rat aortic intima due to transmural pressure.

Huang et al. (1997) propose a new hypothesis and develop a mathematical model to explain rationally the in vitro and in situ measured changes (Tedgui and Lever, 1984; Baldwin and Wilson, 1993) in the hydraulic conductivity of the artery wall of rabbit aorta with transmural pressure. The model leads to the intriguing prediction that this hydraulic conductivity would decrease by one half if the thin intimal layer between the endothelium and the internal elastic lamina volume-compresses approximately fivefold. This paper presents the first measurements of the effect of transmural pressure on intimal layer thickness and shows that the intimal matrix is, indeed, surprisingly compressible. We perfusion-fixed rat thoracic aortas in situ with 2 percent glutaraldehyde solution at 0, 50, 100, or 150 mm Hg lumen pressure and sectioned for light and electron microscopic observations. Electron micrographs show a dramatic, nonlinear decrease in average intimal thickness, i.e., 0.62 +/- 0.26, 0.27 +/- 0.14, 0.15 +/- 0.10, and 0.12 +/- 0.07 (SD) micron for 0, 50, 100, and 150 mm Hg lumen pressure, respectively. The volume strain of the intima is more than 20 times greater than the radial strain of the artery wall due to hoop tension and two orders of magnitude greater than the consolidation of the artery wall as a whole assuming constant medial density (Chuong and Fung, 1984). Moreover, in both light and electron microscopic observations, it is easy to find numerous sites where the endothelium puckers into the fenestral pores at high lumen pressure, as predicted by the theory in Huang et al. (1997). In contrast, the average diameter of a fenestral pore increases only 10 percent as the lumen pressure is increased from 0 to 150 mm Hg. These results indicate that the thin intimal layer comprising less than 1 percent of the wall thickness can have a profound effect on the filtration properties of the wall due to the large change in Darcy permeability of the layer and the large reduction in the entrance area of the flow entering the fenestral pores, though the pores themselves experience only a minor enlargement due to hoop tension.