199 research outputs found
Mathematical Properties of Pump-Leak Models of Cell Volume Control and Electrolyte Balance
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 pump-leak
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 pump-leak models. We treat two cases. When the ion
channel currents have a linear current-voltage 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 pump-leak models, which
can be used as a Lyapunov function to study stability
A Multidomain Model for Ionic Electrodiffusion and Osmosis with an Application to Cortical Spreading Depression
Ionic electrodiffusion and osmotic water flow are central processes in many
physiological systems. We formulate a system of partial differential equations
that governs ion movement and water flow in biological tissue. A salient
feature of this model is that it satisfies a free energy identity, ensuring the
thermodynamic consistency of the model. A numerical scheme is developed for the
model in one spatial dimension and is applied to a model of cortical spreading
depression, a propagating breakdown of ionic and cell volume homeostasis in the
brain.Comment: submitted for publication, Aug. 28, 201
A Model of Electrodiffusion and Osmotic Water Flow and its Energetic Structure
We introduce a model for ionic electrodiffusion and osmotic water flow
through cells and tissues. The model consists of a system of partial
differential equations for ionic concentration and fluid flow with interface
conditions at deforming membrane boundaries. The model satisfies a natural
energy equality, in which the sum of the entropic, elastic and electrostatic
free energies are dissipated through viscous, electrodiffusive and osmotic
flows. We discuss limiting models when certain dimensionless parameters are
small. Finally, we develop a numerical scheme for the one-dimensional case and
present some simple applications of our model to cell volume control
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