Carbon transport revisited: a novel approach for solving quasi-stationary carbon transport in a system with Michaelis-Menten sources and sinks

The Carbon Transport-Resistance Allocation Model (C-TRAM) developed for L-system plant models is based on an analogy between the equations for the osmoticpressure generated phloem-sap flow and Ohm's law. This analogy has proved to be rather confusing because some of the model variables, for example source/sink 'electromotive force', do not have any physiological interpretation. Also, the fact that the transport equations are formulated in terms of phloem-sap flow Jp, while organ growth rates are usually expressed in mass carbohydrate flux Js, does not improve model clarity and consistency. The current paper presents a direct way of solving transport equations based on the Munch hypothesis, consistent with boundary conditions in the form of Michaelis-Menten source/sink fluxes, and accounting for the effects of carbohydrate concentration on phloem resistance. Starting from a system of equations for steady-state coupled phloem/xylem flow (Hall and Minchin, 2013), for a constant water potential in the xylem, an equation for the solute mass flow Js in a conduit element can be obtained in a closed form. The boundary problem in the context of L-system plant models was solved using analytical transformations and computational methods similar to the folding/unfolding algorithm of C-TRAM but not based on the electric circuit analogy. The performance of the transport model was tested using a simple system where exact analytical solutions were available (Hall and Minchin, 2013). To our knowledge, this is the first L-system model that combines a rigorous treatment of the carbohydrate transport in a growing system with Michaelis-Menten source/sink functions. The model will be useful in cases where the actual value of the carbohydrate concentration in phloem is important, e.g., sugar signalling. Currently the model is being extended to take into account phloem/xylem coupling.