Calcium Waves in Differentiated Neuroblastoma Cells
Catherine
Lloyd
Bioengineering Institute, University of Auckland
Model Status
This model is a lumped parameter ODE model produced by taking the original PDE model and stripping out all spatial variation; this gives a spatially homogeneous model which doesn't take into account diffusion.
This model validates with the validation service from the CellML API with no errors or warnings, and has been successfully simulated with the CellML Integration Service using the IDA and CVODE AdamsMoulton order 112 integrators. Due to the change to a lumped parameter formulation, the results of this model are not expected to be comparable to the published paper, nor to any biological reality, and so the results have not been checked against any other source.
Model Structure
Intracellular calcium dynamics are frequently the subject of theoretical mathematical models (De Young and Keizer, 1992, Li and Rinzel, 1994, Keizer and Levine, 1996, JafriRiceWinslow, 1998, and Snyder et al., 2000 are just a few examples of calcium dynamic models that have been coded up into CellML). The physical and chemical laws of calcium waves and oscillations can be expressed in terms of differential equations describing reaction kinetics, fluxes through membranes, and diffusion.
In this study, Charles C. Fink et al. produce an imagebased model of a intracellular calcium wave in differentiated neuroblastoma cells (see below). One important conclusion from their analysis is that neuronal morphology plays a key role in controlling and shaping the inositol1,4,5triphosphate (IP3) dynamics that underlie the calcium wave. The model is comprised of several components including:
IP3 dynamics

which account for IP3 synthesis at the plasma membrane, diffusion into the cytosol, and its degradation.
Calcium dynamics

which calculate the changing intracellular calcium concentration.
Channel kinetics

to describe calcium release from the endoplasmic reticulum (ER) into the cytosol through an IP3sensitive channel.
SERCA pump kinetics

to describe calcium uptake into the ER via the sarcoplasmic endoplasmic reticulum ATPase (SERCA) pumps.
Leak

which models calcium leak from the ER to the cytosol.
and
Calcium buffering

with endogenous buffers.
Their model is based on experimental data. The binding of bradykinin (BK) to its receptor initiates a Gprotein cascade, activation of phospholipase C, and degradation of phosphatidylinositol bisphosphate (PIP2) to IP3. IP3 then diffuses through the cytosol from the plasma membrane to the ER where it activates Ca2+ release through the IP3R channel. The concentration of cytosolic Ca2+ rises, and is subsequently reduced as Ca2+ binds to calcium buffers and is pumped back into the ER through the SERCA. This Ca2+ wave was captured by Fink et al. through the use of fluorescent microscopy. The model of this process was assembled using the Virtual Cell, a computational system for integrating experimentally recorded image, biochemical and electrophysiological data. The model was tested by comparing several simulation results with the real experimental data, and Fink et al. found that there was good spatiotemporal agreement between the two data sets.
It should be noted that the following CellML description (for the raw CellML description of the model, see below) is not quite true to the mathematical model published in the original paper (referenced below). Currently CellML is unable to handle spatial elements, but this will hopefully be possible in the near future with the development of FieldML, an XML based language for spatially variable models. This is important, as the relative positions of the cellular components such as receptors, pumps, channels and enzymes will determine the length of diffusion pathways and therefore the rate of reactions.
An ImageBased Model of Calcium Waves in Differentiated Neuroblastoma Cells, Charles C. Fink, Boris Slepchenko, Ion I. Moraru, James Watras, James C. Schaff, and Leslie M. Loew, 2000,
Biophysical Journal
, 79, 163183. (Full text and PDF versions of the article are available to subscribers on the Biophysical Journal website.) PubMed ID: 10866945
Schematic diagram of model
$\mathrm{j\_IP3}=\mathrm{J\_IP3}e^{\mathrm{k\_0}\mathrm{time}}\frac{d \mathrm{time}}{d \mathrm{time}}=(\mathrm{k\_degr}(\mathrm{IP3}\mathrm{IP3\_0}))$
$\frac{d \mathrm{time}}{d \mathrm{time}}=\mathrm{alpha}(\mathrm{J\_channel}+\mathrm{J\_pump}+\mathrm{J\_leak})+\mathrm{R\_buffering}$
$\mathrm{J\_channel}=\mathrm{J\_max}(\frac{\mathrm{IP3}}{\mathrm{IP3}+\mathrm{K\_IP3}}\frac{\mathrm{Ca}}{\mathrm{Ca}+\mathrm{K\_act}}h)^{3.0}(1.0\frac{\mathrm{Ca}}{\mathrm{Ca\_ER}})\frac{d \mathrm{time}}{d \mathrm{time}}=\mathrm{k\_on}(\mathrm{K\_inh}h(\mathrm{Ca}+\mathrm{K\_inh}))$
$\mathrm{J\_pump}=\mathrm{V\_max}\frac{\mathrm{Ca}^{2.0}}{\mathrm{Ca}^{2.0}+\mathrm{K\_p}^{2.0}}$
$\mathrm{J\_leak}=L(1.0\frac{\mathrm{Ca}}{\mathrm{Ca\_ER}})$
$\mathrm{R\_buffering}=\mathrm{R1}+\mathrm{R2}\mathrm{R1}=(\mathrm{k1\_on}\mathrm{Ca}\mathrm{B1})+\mathrm{k1\_off}\mathrm{CaB1}\mathrm{R2}=(\mathrm{k2\_on}\mathrm{Ca}\mathrm{B2})+\mathrm{k2\_off}\mathrm{CaB2}\frac{d \mathrm{time}}{d \mathrm{time}}=\mathrm{R1}\frac{d \mathrm{time}}{d \mathrm{time}}=\mathrm{R1}\frac{d \mathrm{time}}{d \mathrm{time}}=\mathrm{R2}\frac{d \mathrm{time}}{d \mathrm{time}}=\mathrm{R2}\mathrm{K1}=\frac{\mathrm{k1\_off}}{\mathrm{k1\_on}}\mathrm{K2}=\frac{\mathrm{k2\_off}}{\mathrm{k2\_on}}$
$\mathrm{gamma}=\begin{cases}\mathrm{gamma\_s} & \text{if $\mathrm{soma\_or\_neurite}\le 0$}\\ \mathrm{gamma\_n} & \text{otherwise}\end{cases}\mathrm{gamma\_s}=\frac{\mathrm{gamma\_0}\mathrm{sigma}}{\mathrm{delta}\mathrm{sigma\_neurite\_2D}\mathrm{w\_n}+\mathrm{sigma\_soma\_2D}\mathrm{w\_s}}\mathrm{gamma\_n}=\frac{\mathrm{gamma\_0}\mathrm{sigma}\mathrm{delta}}{\mathrm{delta}\mathrm{sigma\_neurite\_2D}\mathrm{w\_n}+\mathrm{sigma\_soma\_2D}\mathrm{w\_s}}\mathrm{j\_Ca}=\begin{cases}\mathrm{gamma}(\mathrm{Ca}\mathrm{Ca\_c}) & \text{if $\mathrm{Ca}> \mathrm{Ca\_c}$}\\ 0.0 & \text{otherwise}\end{cases}$
Neuroblastoma
calcium dynamics
Catherine
Lloyd
May
20020729
An imagebased model of calcium waves in differentiated
neuroblastoma cells
79
163
183
Catherine Lloyd
10866945
James
Schaff
C
c.lloyd@auckland.ac.nz
Corrected equations: removed all lapacian functions  spatial elements cannot be handled by CellML 1.0.
This is the CellML description of Fink et al's 2000 model of calcium
waves in differentiated neuroblastoma cells.
It should be noted that the following CellML description is not quite
true to the mathematical model published in the original paper
referenced below. Currently CellML is unable to handle spatial
elements, but this will hopefully be possible in the near future with
the development of FieldML.
Catherine
Lloyd
May
Catherine
Lloyd
May
Corrected equations: IP3_diff_eq in IP3_dynamics, Ca_diff_eq in
Calcium_dynamics, and B2_diff_eq and CaB2_diff_eq in
Calcium_buffering.
Boris
Slepchenko
The University of Auckland, Bioengineering Institute
20030604
200007
Charles
Fink
C
The University of Auckland
The Bioengineering Institute
keyword
Biophysical Journal
Ion
Moraru
I
Leslie
Loew
M
James
Watras
Fink et al's 2000 model of calcium waves in differentiated neuroblastoma cells.
Neuroblastoma
20030604