A Bifurcation Analysis of Two Coupled Calcium Oscillators
Catherine
Lloyd
Auckland Bioengineering Institute, The University of Auckland
Model Status
This CellML model runs in both OpenCell and COR to produce an oscillating output similar to that from the original published model. The units have been checked and they are consistent.
Model Structure
ABSTRACT: In many cell types, asynchronous or synchronous oscillations in the concentration of intracellular free calcium occur in adjacent cells that are coupled by gap junctions. Such oscillations are believed to underlie oscillatory intercellular calcium waves in some cell types, and thus it is important to understand how they occur and are modified by intercellular coupling. Using a previous model of intracellular calcium oscillations in pancreatic acinar cells, this article explores the effects of coupling two cells with a simple linear diffusion term. Depending on the concentration of a signal molecule, inositol (1,4,5)-trisphosphate, coupling two identical cells by diffusion can give rise to synchronized in-phase oscillations, as well as different-amplitude in-phase oscillations and same-amplitude antiphase oscillations. Coupling two nonidentical cells leads to more complex behaviors such as cascades of period doubling and multiply periodic solutions. This study is a first step towards understanding the role and significance of the diffusion of calcium through gap junctions in the coordination of oscillatory calcium waves in a variety of cell types.
The original paper reference is cited below:
A bifurcation analysis of two coupled calcium oscillators, Michael Bindschadler and James Sneyd, 2001, CHAOS, 11, 237-246. PubMed ID: 12779457
cell diagram
Schematic diagram of the IP3 receptor model. The receptor with its three possible states: X, Y, and Z; representing open, shut and inactive respectively, is embedded within a model of intracellular calcium dynamics.
h1
receptor concentration in cell 1
$\frac{d \mathrm{h1}}{d \mathrm{time}}=\mathrm{phi3\_c1}(1-\mathrm{h1})-\frac{\mathrm{phi1\_c1}\mathrm{phi2\_c1}\mathrm{h1}p}{\mathrm{phi1\_c1}p+\mathrm{phi\_1\_c1}}$
h2
receptor concentration in cell 2
$\frac{d \mathrm{h2}}{d \mathrm{time}}=\mathrm{phi3\_c2}(1-\mathrm{h2})-\frac{\mathrm{phi1\_c2}\mathrm{phi2\_c2}\mathrm{h2}p}{\mathrm{phi1\_c2}p+\mathrm{phi\_1\_c2}}$
phi
rate functions
$\mathrm{phi1\_c1}=\frac{\mathrm{r2}\mathrm{c1}}{\mathrm{R1}+\mathrm{c1}}\mathrm{phi\_1\_c1}=\frac{\mathrm{k1}}{\mathrm{R3}+\mathrm{c1}}\mathrm{phi2\_c1}=\frac{\mathrm{k2}+\mathrm{r4}\mathrm{c1}}{\mathrm{R3}+\mathrm{c1}}\mathrm{phi3\_c1}=\frac{\mathrm{k3}}{\mathrm{R5}+\mathrm{c1}}\mathrm{phi1\_c2}=\frac{\mathrm{r2}\mathrm{c2}}{\mathrm{R1}+\mathrm{c2}}\mathrm{phi\_1\_c2}=\frac{\mathrm{k1}}{\mathrm{R3}+\mathrm{c2}}\mathrm{phi2\_c2}=\frac{\mathrm{k2}+\mathrm{r4}\mathrm{c2}}{\mathrm{R3}+\mathrm{c2}}\mathrm{phi3\_c2}=\frac{\mathrm{k3}}{\mathrm{R5}+\mathrm{c2}}$
j_pump
Ca2+ ATPase pump
$\mathrm{j\_pump\_c1}=\frac{\mathrm{Vp}\mathrm{c1}^{2}}{\mathrm{Kp}^{2}+\mathrm{c1}^{2}}$
$\mathrm{j\_pump\_c2}=\frac{\mathrm{Vp}\mathrm{c2}^{2}}{\mathrm{Kp}^{2}+\mathrm{c2}^{2}}$
j_receptor
IP3 receptor
$\mathrm{j\_receptor\_c1}=\mathrm{kf}\left(\frac{p\mathrm{h1}\mathrm{phi1\_c1}}{\mathrm{phi1\_c1}p+\mathrm{phi\_1\_c1}}\right)^{4}$
$\mathrm{j\_receptor\_c2}=\mathrm{kf}\left(\frac{p\mathrm{h2}\mathrm{phi1\_c2}}{\mathrm{phi1\_c2}p+\mathrm{phi\_1\_c2}}\right)^{4}$
j_diffusion
calcium diffusion
$\mathrm{j\_diffusion}=D(\mathrm{c2}-\mathrm{c1})$
c1
calcium concentration in cell 1
$\frac{d \mathrm{c1}}{d \mathrm{time}}=\mathrm{j\_receptor\_c1}-\mathrm{j\_pump\_c1}+\mathrm{j\_leak}+\mathrm{j\_diffusion}$
c2
calcium concentration in cell 2
$\frac{d \mathrm{c2}}{d \mathrm{time}}=\mathrm{j\_receptor\_c2}-\mathrm{j\_pump\_c2}+\mathrm{j\_leak}+\mathrm{j\_diffusion}$
bifurcation
oscillator
calcium dynamics
A Bifurcation Analysis of Two Coupled Calcium Oscillators
11
237
246
2007-12-06T14:07:34+13:00
2004-03-14T00:00:00+00:00
12779457
James Lawson
This model has been curated and is known to run in PCEnv. The model is able to produce oscillating output similar to that shown in figures from the publication.
c.lloyd@auckland.ac.nz
keyword
James
Lawson
Richard
Catherine Lloyd
The University of Auckland, Auckland Bioengineering Institute
Catherine
Lloyd
May
Michael
Bindschadler
2009-05-28T14:54:14+12:00
Vignesh
Kumar
This is the CellML description of Bindschadler and Sneyd's bifurcation analysis of two coupled calcium oscillators.
Bindschadler and Sneyd's bifurcation analysis of two coupled calcium oscillators.
Chaos
The University of Auckland
Auckland Bioengineering Institute
James
Sneyd
Fixed duplicate connections
Fixed CellML 1.0/1.1 namespace mixing
Updated documentation, curation status
50
100000
0.001
2001-03
Recoded model using parameters used by Biomodels Database as a guide