| Abstract | Cell volume and concentration regulation are universal in biology, and examples of active and passive transport of water and solutes across the cell membrane can be drawn from all areas of both plant and animal biology. Recently a general model encompassing multiple transport phenomena for cell volume and concentration regulation of water and a single solute with applications in several areas of biology was published and a local stability result was shown (Hernandez 2007). Here, we extend this model to the case with water and multiple solutes and solute pathways (e.g.~both active and passive transport), and extend the previous stability result by proving global asymptotic stability at the rest point. Because it is often useful in biology to understand the optimal way to modify the concentration of intracellular solutes, we use this stability result together with some classic results from optimal control theory to show that the system is controllable and an optimal control exists in the case of multiple solutes or solute pathways. Finally we determine explicitly the optimal control protocol (that is, synthesize the time optimal control problem) for the commonly encountered water-single permeating and single non-permeating solute case. |
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