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J Math Biol. 2011 Aug;63(2):339-59. doi: 10.1007/s00285-010-0374-4. Epub 2010 Nov 10.

A general model for the dynamics of cell volume, global stability, and optimal control.

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Mathematical and Computational Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD, USA.


Cell volume and concentration regulation in the presence of changing extracellular environments has been studied for centuries, and recently a general nondimensional model was introduced that encompassed solute and solvent transmembrane flux for a wide variety of solutes and flux mechanisms. Moreover, in many biological applications it is of considerable interest to understand optimal controls for both volume and solute concentrations. Here we examine a natural extension of this general model to an arbitrary number of solutes or solute pathways, show that this system is globally asymptotically stable and controllable, define necessary conditions for time-optimal controls in the arbitrary-solute case, and using a theorem of Boltyanski prove sufficient conditions for these controls in the commonly encountered two-solute case.

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