Finite transform solution of the vibrating annular membrane.
Vibrating annular membrane problem, including load per unit area and asymmetry of load and vibration, using finite transform derivation
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Vibrating annular membrane problem, including load per unit area and asymmetry of load and vibration, using finite transform derivation
Introduction. The higher plant is a heterogeneous, mechanically prestressed structure continually subject to shifting forces. When a cell grows in a plant at gravitropic equilibrium, it must create localized maxima of shear in walls of neighboring cells. Such mechanical stress and strain are likely detected in a variety of ways. However, tension-sensitive ion channels are of particular interest because it appears that they are elaborately evolved for sensory function. We hypothesize that 1) the patchy patterns of high shear are focused via wall-to-membrane linkers onto the plasma membrane, where 2) they are translated by mechanosensory cation channels into corresponding patterns of high cytosolic Ca2+, which 3) initiate local enhancement of wall expansion. Further, we hypothesize that the local promotion of enhancement is achieved at least in part by local intensification of auxin transport across the plasma membrane. By implication, when an organ is asymmetrically pressed, rubbed, or bent or when it is displaced in the gravitational field, the net asymmetry of shear stress occurring across the organ would lead to asymmetric redistribution of auxin and corrective asymmetric growth. We shall describe a representative mechanosensitive Ca(2+) -selective cation channel (MCaC) with susceptibilities to xenobiotics implicating it as a force transducer in thigmo- and gravitropism. Then, we shall consider whether a putative wall-to-membrane linker (WML) could be a key feature of the molecular architecture permitting the stress distributed in the wall system to be focused on the channels.
Seven day old etiolated pea epicotyls were loaded symmetrically with 3H-indole 3-acetic acid (IAA) or 45Ca2+, then subjected to 1.5 hours of 1g gravistimulation. Epidermal peels taken from top and bottom surfaces after 90 minutes showed an increase in IAA on the lower side and of Ca2+ on the upper side. Inhibitors of IAA movement (TIBA, 9-hydroxyfluorene carboxylic acid) block the development of both IAA and Ca2+ asymmetries, but substances known to interfere with normal Ca2+ transport (nitrendipine, nisoldipine, Bay K 8644, A 23187) do not significantly alter either IAA or Ca2+ asymmetries. These substances, however, are active in modifying both Ca2+ uptake and efflux through oat and pea leaf protoplast membranes. We conclude that the 45Ca2+ fed to pea epicotyls occurs largely in the cell wall, and that auxin movement is primary and Ca2+ movement secondary in gravitropism. We hypothesize that apoplastic Ca2+ changes during graviresponse because it is displaced by H+ secreted through auxin-induced proton release. This proposed mechanism is supported by localized pH experiments, in which filter paper soaked in various buffers was applied to one side of a carborundum-abraded epicotyls. Buffer at pH 3 increases calcium loss from the side to which it is applied, whereas pH 7 buffer decreases it. Moreover, 10 micromolar IAA and 1 micromolar fusicoccin, which promote H+ efflux, increase Ca2+ release from pea epicotyl segments, whereas cycloheximide, which inhibits H+ efflux, has the reverse effect. We suggest that Ca2+ does not redistribute actively during gravitropism: the asymmetry arises because of its release from the wall adjacent to the region of high IAA concentration, proton secretion, and growth. Thus, the asymmetric distribution of Ca2+ appears to be a consequence of growth stimulation, not a critical step in the early phase of the graviresponse.
Touch and gravity are two of the many stimuli that plants must integrate to generate an appropriate growth response. Due to the mechanical nature of both of these signals, shared signal transduction elements could well form the basis of the cross-talk between these two sensory systems. However, touch stimulation must elicit signaling events across the plasma membrane whereas gravity sensing is thought to represent transformation of an internal force, amyloplast sedimentation, to signal transduction events. In addition, factors such as turgor pressure and presence of the cell wall may also place unique constraints on these plant mechanosensory systems. Even so, the candidate signal transduction elements in both plant touch and gravity sensing, changes in Ca2+, pH and membrane potential, do mirror the known ionic basis of signaling in animal mechanosensory cells. Distinct spatial and temporal signatures of Ca2+ ions may encode information about the different mechanosignaling stimuli. Signals such as Ca2+ waves or action potentials may also rapidly transfer information perceived in one cell throughout a tissue or organ leading to the systemic reactions characteristic of plant touch and gravity responses. Longer-term growth responses are likely sustained via changes in gene expression and asymmetries in compounds such as inositol-1,4,5-triphosphate (IP3) and calmodulin. Thus, it seems likely that plant mechanoperception involves both spatial and temporal encoding of information at all levels, from the cell to the whole plant. Defining this patterning will be a critical step towards understanding how plants integrate information from multiple mechanical stimuli to an appropriate growth response.
Gravity-induced changes in intracellular potentials in primary roots of 2-day-old mung bean (Vigna mungo L. cv. black matpe) seedlings were investigated using glass microelectrodes held by 3-dimensional hydraulic micro-drives. The electrodes were inserted into outer cortical cells within the elongation zone. Intracellular potentials, angle of root orientation with respect to gravity, and position within the root of the impaled cortical cell were measured simultaneously. Gravistimulation caused intracellular potential changes in cortical cells of the elongation zone. When the roots were oriented vertically, the intracellular potentials of the outer cortical cells (2 mm behind the root apex) were approximately - 115 mV. When the roots were placed horizontally cortical cells on the upper side hyperpolarized to - 154 mV within 30 s while cortical cells on the lower side depolarized to about - 62 mV. This electrical asymmetry did not occur in cells of the maturation zone. Because attempts to insert the electrode into cells of the root cap were unsuccessful, these cells were not measured. The hyperpolarization of cortical cells on the upper side was greatly reduced upon application of N,N'-dicyclohexylcarbodiimide (DCCD), an inhibitor of respiratory energy coupling. When stimulated roots were returned to the vertical, the degree of hyperpolarization of cortical cells on the previous upper side decreased within 30 s and approached that of cortical cells in non-stimulated roots. This cycle of hyperpolarization/loss of hyperpolarization was repeatable at least ten times by alternately turning the root from the vertical to the horizontal and back again. The very short (<30 s) lag period of these electrical changes indicates that they may result from stimulus-perception and transduction within the elongation zone rather than from transmission of a signal from the root cap.