Abstract
Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pumpleak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pumpleak models. We treat two cases. When the ion channel currents have a linear currentvoltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pumpleak models, which can be used as a Lyapunov function to study stability.
Original language  English (US) 

Pages (fromto)  873916 
Number of pages  44 
Journal  Journal of Mathematical Biology 
Volume  64 
Issue number  5 
DOIs 

State  Published  Apr 2012 
Externally published  Yes 
Keywords
 Cell volume control
 Differential algebraic system
 Electrolyte balance
 Free energy
 Lyapunov function