Concentration-dependent unloading as a necessary assumption for a closed form mathematical model of osmotically driven pressure flow in phloem

Previous attempts to model steady state Münch pressure flow in phloem (Christy and Ferrier. [1973]. Plant Physiol. 52: 531-538; and Ferrier et al. [1974]. Plant Physiol. 54: 589-600) lack sufficient equations, and results were produced which do not represent correct mathematical solutions. Additional equations for the present closed form model were derived by assuming that unloading of a given solute is dependent upon the concentration of that solute in the sieve tube elements. Examples of linear and enzymic type unloading mechanisms are given, although other concentration-dependent mechanisms could be substituted. A method for a numerical solution is outlined, and proof of convergence is presented along with some representative data and the speed of computer calculations. The model provides the minimal set of equations for describing the Münch pressure flow hypothesis as it might operate in plants.