Equations have been developed to describe the diffusional movement of a weak acid such as the auxin indoleacetic acid through a long file of vacuolated cells, where cellular accumulation is driven by the pH gradients across the cell membranes. If the permeability to the auxin anion is greater at one end of the cell than at the other, diffusional movement takes the form of polar transport, which exhibits: a nearly constant velocity either for the front or for a pulse of radioactive auxin, the capacity to move auxin against an external gradient of concentration, and a polar ratio that increases exponentially with the length of the section. The determinants of velocity include both diffusion through the vacuole and permeation steps at the cell membranes. Except Since the early quantitative descriptions of the transport of the endogenous plant growth hormone auxin* (1-3) its cellular basis has been elusive. Among its characteristics are (i) a polarity, in which auxin moves more effectively through tissues in one direction than in the other, with the polar ratio increasing approximately exponentially with distance (4-6); (ii) the ability to move auxin through a tissue against an external concentration gradient (3); and (iii) a velocity, evidenced by a nearly constant rate of travel of 10 or more mm hr-' of either the front (2, 4, 7-10) or a pulse of radioactively labeled auxin introduced at the apical end of a section (11). Various models involving differential secretion from the basal ends of the cells can account for the first two of these features, but there has been no convincing explanation for the third.The chemosmotic polar diffusion hypothesis draws ideas and observations from several sources and postulates that polar transport involves both steps of membrane permeation and diffusion through the cell (12). Uptake of auxin is pH dependent and appears to be driven by the pH difference between the inside of the cell and the acidic wall space. With a difference of 2 pH units, and with only the undissociated acid permeant, auxin can accumulate passively to an internal concentration more than 50 times greater than the external. If the auxin anion is also permeant, this accumulation will be reduced. The suggestion that the polarity of transport is caused by a greater permeability to the anion at the basal than at the apical end of each cell (13, 14) is a central feature of this hypothesis.Because neither cytoplasmic streaming (15) nor a lateral sheath of cytoplasm around the vacuole (16) is necessary to support transport, the auxin seems to cross the tonoplast and diffuse through the vacuole in traversing the length of each cell. The proposed path of auxin movement (Fig. 1) The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.

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In this note the problem of determining the propagation of a plane shock moving through a polytropic gas into an undisturbed body of the gas, leaving a non-isentropic disturbance behind it, is reduced to the solution of the problem of Cauchy for a Monge-Ampére partial differential equation of special type. The application of this result to specific examples is being studied with a view to later publication.

We present here explicit mathematical formulas for calculating the concentration, mass, and velocity of movement of the center of mass of the plant growth regulator auxin during its polar movement through a linear file of cells. The results of numerical computations for two cases, (a) the conservative, in which the mass in the system remains constant and (b) the non-conservative, in which the system acquires mass at one end and loses it at the other, are graphically presented. Our approach differs from that of Mitchison's (Mitchison 1980) in considering both initial effects of loading and end effects of substance leaving the file of cells. We find the velocity varies greatly as mass is entering or leaving the file of cells but remains constant as long as most of the mass is within the cells. This is also the time for which Mitchison's formula for the velocity, which neglects end effects, reflects the true velocity of auxin movement. Finally, the predictions of the model are compared with two sets of experimental data. Movement of a pulse of auxin through corn coleoptiles is well described by the theory. Movement of auxin through zucchini shoots, however, shows the need to take into account immobilization of auxin by this tissue during the course of transport.