Intercellular spiral waves of calcium
Catherine
Lloyd
Auckland Bioengineering Institute, The University of Auckland
Model Status
This model contains partial differentials and as such can not currently be solved by existing CellML tools.
Model Structure
ABSTRACT: Intercellular calcium waves have been observed in a large number of cell types, and are known to result from a variety of stimuli, including mechanical or hormonal stimulation. Recently, spiral intercellular waves of calcium have been observed in slices of hippocampal tissue. We use an existing model to study the properties of spiral intercellular calcium waves. Although intercellular spiral waves are well known in the context of cardiac muscle, due to the small value of the calcium diffusion coefficient intercellular calcium waves have fundamentally different properties. We show that homogenisation techniques give a good estimate for the plane wave speed, but do not describe spiral behaviour well. Using an expression for the effective diffusion coefficient we estimate the intercellular calcium permeability in liver. For the bistable equation, we derive an analytic estimate for the value of the intercellular permeability at which wave propagation fails. In the calcium wave model, we show numerically that the spiral period is first a decreasing, then an increasing, function of the intercellular permeability. We hypothesise that this is because the curvature of the spiral core is unimportant at low permeability, the period being approximately set instead by the speed of a plane wave along a line of coupled cells in one dimension.
The original paper reference is cited below:
Intercellular Spiral Waves of Calcium, Matthew Wilkins and James Sneyd, 1998,Journal of Theoretical Biology, 191, 299-308. PubMed ID: 9631570
reaction diagram
Schematic diagram showing the transmembrane calcium fluxes which are described by the mathematical model.
$\frac{\partial^{1}c}{\partial \mathrm{time}}=\mathrm{Dc}(\mathrm{dc\_dx2}+\mathrm{dc\_dy2})+\mathrm{j\_flux}-\mathrm{j\_pump}\frac{\partial^{2}c}{\partial x^{2.0}}=\mathrm{dc\_dx2}\frac{\partial^{2}c}{\partial y^{2.0}}=\mathrm{dc\_dy2}$
$\mathrm{j\_pump}=\frac{\mathrm{gamma}c}{\mathrm{k\_gamma}+c}$
$\mathrm{j\_flux}=\mathrm{kf}\mathrm{mu\_p}(b+\frac{(1.0-b)c}{\mathrm{k1}+c})h\frac{d h}{d \mathrm{time}}=\frac{\frac{\mathrm{k2}^{2.0}}{\mathrm{k2}^{2.0}+c^{2.0}}-h}{\mathrm{tau\_h}}\frac{d \mathrm{mu\_p}}{d \mathrm{time}}=\frac{p}{\mathrm{k\_mu}+p}$
calcium dynamics
calcium waves
ca spiral waves
Auckland Bioengineering Institute, The University of Auckland
James
Sneyd
9631570
This CellML model has been checked, but since the published model involves partial differential equations it requires FieldML [or other applicaiton] in order to operate.
2008-06-03T12:02:38+12:00
The University of Auckland
Auckland Bioengineering Institute
Geoff Nunns
Catherine Lloyd
c.lloyd@auckland.ac.nz
Wilkins and Sneyd's 1998 mathematical model of intercellular spiral waves of calcium.
1998-04-07
I attempted to curate this model, which had been imported from the previous database, but it contained partial differential equations and could not be solved in CellML.
2004-03-20T00:00:00+00:00
Catherine
Lloyd
May
keyword
This is the CellML description of Wilkins and Sneyd's 1998 mathematical model of intercellular spiral waves of calcium.
Matthew
Wilkins
Intercellular Spiral Waves of Calcium
191
299
308
Journal of Theoretical Biology
Geoffrey
Nunns