Intracellular Calcium Dynamics in Endothelial Cells
Scott Cara; Anita Layton
Department of Mathematics, Duke University
Abstract
Utilizing the models presented by Edwards et al. (2008) and Higgins et al. (2006),
I have formulated a mathematical model for the changing concentration of
calcium within an individual cell. The model is based upon an endothelial cell
found in the descending vasa recta (DVR) in the kidney; however, one of the
major goals of this model is to be just as effective in other cells outside the DVR
given proper parameters and steady state conditions. The model is made up of
over fifty differential equations representing membrane potential, changes in ionic
concentrations, buffering kinetics, and channel opening probabilities. The cell is
divided into three compartments for analysis: the cytoplasm, the microdomain,
and the endoplasmic reticulum. The time derivatives are monitored through ionic
currents and voltage potentials based on the model by Edwards et al. The
SERCA pump, found in the membrane between the cytoplasm and endoplasmic
reticulum, is modeled after Higgins et al., in which its is presented as a kinetic
pump. The model shows that slight alterations in extracellular potassium
concentrations, IP3 concentrations, inhibition of ionic channels at the cell
membrane or the endoplasmic reticulum membrane, or other parameter changes
can lead to calcium spikes. These spikes can either be oscillating or singular
based upon the parameter variation. Furthermore, this model allows the cell to
obtain no futile cycling at rest with the endoplasmic reticulum. This phenomenon,
in which there is no current across the endoplasmic reticulum membrane in either
direction, is in direct causality of using the bidirectional, kinetic SERCA pump as
given by Higgins et al. and has been precluded to in several studies as described
by Higgins et al.
Model
The non-kinetic SERCA pump is given by a Hill Equation with coefficient 2.
The ionic currents in the model are examined for potassium, sodium, chloride, and calcium.
The ionic currents are modeled after Edwards et al. (2008). The model contains gates for IP3
and Ryanodine release, also modeled after Edwards et al. These gates are assumed to be
directly proportional to some probability of being opened, with the ryanodine probability being
proportional to the concentration of calcium in the compartment (microdomain or cytoplasm).
The model also assumes some flux of ions between the two compartments and is described
using the model in Edwards et al.
The SERCA pumps are modeled both non-kinetically, as prescribed by Edwards et al. and
kinetically, as described by Higgins et al. (2006). The buffering effect of proteins in the cell can
be turned on or off, given that experimental data has shown that some cells have this effect
(including the cells in the DVR), while others do not. Buffers are given by three different
proteins based upon experimental findings.
The endoplasmic reticulum is modeled
two different ways:
1. As a single large store of calcium
with both the IP3 and Ryanodine
attached to one large pool (A).
2.As two large stores of calcium
one large compartment),
containing all the the IP3
one for all of the
gates
(in
one
movement and
ryanodine receptor (B).
Calcium is the fifth most abundant element in the human body. The human body
requires calcium in such large quantities because calcium serves an important role
as an intercellular messenger for neurons and muscle cells as well as a catalyst
for enzymes to work properly. For these reasons, the importance of a wellregulated level of calcium inside cells is critical for proper bodily function. Thus,
much work has been done to properly model the dynamics of calcium
concentrations within individual cells.
Edwards et al. (2008) presents a model of the cell that takes into effect this
experimentally recognized calcium induced-calcium release. The model expresses
two major sources of release from the endoplasmic reticulum: the inositol
trisphosphate (IP3) receptor and the ryanodine (Ry) receptor. Through a slight
change in some parameter in the cell, this system can create large releases of
calcium from the endoplasmic reticulum into the cytoplasm and microdomain. To
restore normal concentrations, the model presumes a Sarco(endo)plamsic
reticulum Ca-ATPase (SERCA) pump, which pumps calcium back into the
concentrated store. Higgins et al. (2006) presents a kinetic model of this pump.
Calcium
Oscillations
While acting as a cellular messenger to perform
some task, it is often crucial for the cell to produce calcium oscillations to repeat a task while
some effect is occurring. Using the non-kinetic model of the SERCA pump and a different store
endoplasmic reticulum while allowing for a buffering effect, we can see this calcium oscillation
by raising the IP3 production in the cell by 100% and inhibiting potassium ion currents by 90%.
Below is a flow chart of the actions that occur to produce oscillations followed by a graph of the
results.
IP3R-induced Leakage into
Cytoplasm and VoltageInduced ionic current leakage
of calcium into cytoplasm
Cell reaches reaches near
steady state, IP3 production is
still elevated
Calcium induced-calcium
release from Ryanodine
Receptor into cytoplasm
NCX (sodium-calcium) currents
increase carrying calcium back
out of cell and SERCA pumps
increase carrying calcium back
into ER
Below are the effects of buffering on the model using a kinetic SERCA pump by
inducing KCl increase in the extracellular environment. Once again, the left shows
common store and the right shows different store.
COMMON STORE
Where ISERCA,max is the maximum flow through the SERCA pump, fj is the fraction that the
compartment (cytoplasm or microdomain makes up of the cell), Kmf and Kmr are constants,
[Ca]j is the calcium concentration in compartment j (cytoplasm or microdomain) and [Ca]SR is
the calcium concentration in the endoplasmic reticulum.
In contrast, the kinetic SERCA pump is found by letting the speed of the SERCA
pump tend to infinite so that the pump can work as fast as necessary to return to steady
state if there is a disruption in calcium concentration. Then, this is given by:
DIFFERENT STORE
in
Introduction
Edwards et al. (2008) analyzed the cell by dividing the model into three
dynamically changing compartments: the regular cytoplasm, a key pocket of
cytoplasm known as the microdomain, and the endo(sarco)plasmic reticulum
[Sarcoplamsic reticulum is a type of endoplasmic reticulum found in muscles and
the model uses these names interchangeable]. The endoplasmic reticulum is a
large store of calcium found in the cell that comprises nearly an eighth of the cell’s
volume. It is commonly known that when the calcium ion acts as a messenger
between cells, it does so by causing large spikes in calcium concentration in the
cytoplasm by endoplasmic release followed by a restoration to normal
concentration levels. The mechanism for this release is referred to as calcium
induced-calcium release, where a small increase in calcium concentration will lead
to a large spike. The method of this mechanism and the act of restoring calcium
levels to pre-spike concentrations is the basis of my model.
Kinetic Pump – Buffering Effect
Kinetic vs. Non-Kinetic SERCA Pump
Where K12, K32, k-2, k-4, k2, and k4 are constants, Pt is the total concentration of pump protein
present (assumed as 15mM/mol for all models), ce is the concentration of calcium in the
endoplasmic reticulum, and c is the concentration of calcium in the given compartment
(microdomain or cytoplasm).
Although these two SERCA pump
models are both, the kinetic pump has
the ability to work much faster.
This is key in returning the calcium
levels to pre-disruption steady states
and has a large effect on the model
as a whole. As such, the kinetic pump
can cause higher calcium spikes.
At first, the pump is nearly off but when
it senses high outside calcium
concentration it is able to immediately
pump the calcium back to normal levels.
This effect allows for higher and sharper
peaks. For example, on the right is the
effect of KCl addition to the extracellular
environment for a common store kinetic
and non-kinetic pump. Notice that the
non-kinetic peak is almost two orders of
magnitude less than the peak of the
kinetic pump.
Non-Kinetic Pump – Buffering Effect
As suggested by Higgins et al., the results of adding buffers to a model are a significantly
lower amplitude and a slightly increased period (if oscillations) or a less abrupt peak. In the
non-kinetic model, you can see this effect on cytoplasmic calcium concentration by adding
KCl to the extracellular environment to induce calcium induced-calcium release. The figure
below shows this effect [Left is common store, right is different store].
COMMON STORE
DIFFERENT STORE
It is clear from this that the buffering effect in the model has a large effect on the
amplitude and length of spikes. From the different store model, we can see that both
the peaks and oscillations are more spread out over time given buffering. Also, the
amplitude changes are as follows:
From this data, we can see that, although being in different or common store
endoplasmic reticulum models has little overall effect on how the buffering effect
changes the model, the buffering effect has a large difference based upon the
kineticity of the SERCA pump. This is also evident in the increasing period or time
length of calcium spikes seen in the graphs.
Summary
1. This model is a single cell interpretation of the calcium dynamics occurring given
varying parameters in the endothelial cell of the DVR. It accurately corresponds to
experimental data on the subject.
2. A variation in the potassium concentration, inhibition of potassium channels coupled
with IP3 production increase, and other changes can lead to calcium induced –calcium
release spikes within the cell. These spikes can either oscillate or return to a steady
state condition dependent upon the parameters.
3. It is possible to achieve zero futile cycling at rest between the cytoplasm (and
microdomain) and the endoplasmic reticulum with this model. Experimentally and
theoretically, this event is likely given that the cell does not usually want to waste
energy fruitlessly.
4. The buffering effect of considering protein-bound calcium ions in the cell is much more
prominent when modeling using a kinetic SERCA pump rather than a non-kinetic
SERCA pump due to the increased speed of the kinetic SERCA pump.
1.
References
Higgins, Erin R., Mark B. Cannell, and James Sneyd. "A Buffering SERCA
Pump in Models of Calcium Dynamics." Biophysical Journal 91.1 (2006): 15163.
2. Edwards, Aurélie, and T. L. Pallone. "Mechanisms Underlying Angiotensin IIinduced Calcium Dynamics." American Journal of Physiology (2008)
295(2):F568-84.
3. Edwards, Aurélie, and T. L. Pallone. “Modification of cytosolic calcium signaling
by subplasmalemmal microdomains.” American Journal of Physiology (2006)
292(6):F1827-F1845.