The pacemaker activity of interstitial cells of Cajal (ICCs) has been known to initiate the propagation of slow waves along the whole gastrointestinal tract through spontaneous and repetitive generation of action potentials. We studied the mechanism of the pacemaker activity of ICCs in the mouse small intestine and tested it using a mathematical model. The model includes ion channels, exchanger, pumps and intracellular machinery for Ca2+ regulation. The model also incorporates inositol 1,4,5-triphosphate (IP3) production and IP3-mediated Ca2+ release activities. Most of the parameters were obtained from the literature and were modified to fit the experimental results of ICCs from mouse small intestine. We were then able to compose a mathematical model that simulates the pacemaker activity of ICCs. The model generates pacemaker potentials regularly and repetitively as long as the simulation continues. The frequency was set at 20 min−1 and the duration at 50% repolarization was 639 ms. The resting and overshoot potentials were −78 and +1.2 mV, respectively. The reconstructed pacemaker potentials closely matched those obtained from animal experiments. The model supports the idea that cyclic changes in [Ca2+]i and [IP3] play key roles in the generation of ICC pacemaker activity in the mouse small intestine.
Rhythmical contractions of the gastrointestinal (GI) tract are associated with the pacemaker electrical activity generated in the muscle layers, which occurs at low frequency in the absence of an extrinsic nervous stimulation. The ICCs, distributed in the myenteric region of the GI wall, have been suggested to initiate the pacemaker activity (Tomita 1981), which then propagates to the smooth muscle cells through gap junctions to generate the contraction of the whole GI tract (Dickens et al. 1999). This pacemaker activity is represented by the driving potential or pacemaker potential of ICC. However, the mechanism of the pacemaker activity is not yet well understood.
Recently, Goto et al. (2004) recorded the pacemaker activity in a single ICC prepared from mouse small intestine. The amplitude of the pacemaker potential was slightly above 70 mV and a series of voltage-clamp experiments was performed to investigate the mechanism of the pacemaker activity. They recorded a large inward current with an autonomous time course (IAI) by applying a depolarizing pulse and concluded that the spontaneous depolarization is caused by the activation of the IAI. Now, it is possible to develop a mathematical model to explain the regenerative nature of pacemaker potentials in a single ICC. Although some investigators have made mathematical models to simulate the regenerative potentials in the GI tract (Miftakhov et al. 1999; Edwards & Hirst 2003, 2005), the models are still phenomenological in the point of method or approach, and they focused on the electrical activity of a larger cellular network rather than a single ICC.
In this study, pacemaker potentials of ICCs from the mouse small intestine were simulated with a mathematical model to explain the regenerative nature of the pacemaker potentials and the underlying [Ca2+]i changes. The model faithfully reproduces the pacemaker activity and suggests a possible mechanism.
Modelling of the pacemaker activity in the ICCs is still at a very early stage in comparison with modelling of the pacemaker activity of the heart. Furthermore, the experimental results describing the Ca2+ dynamics, ion channels and intracellular metabolic pathways in ICCs are scarce. Thus, although our object is to make a simulation model of mouse ICCs, the equations and parameters used were partly derived from the heart models (Luo & Rudy 1994; Matsuoka et al. 2003) and were modified to reproduce the pacemaker activity of mouse ICCs (table 1).
(a) Cellular geometry
The ICCs isolated from mouse small intestine have a spindle-shaped cell body with extending processes. The cell body is 5–15 μm wide, and the measured mean capacitance is 21–25 pF (Kim et al. 2002; Koh et al. 2002; Goto et al. 2004). The cell volume is simply calculated from a cell capacitance of 25 pF by assuming that the cell has the same hexahedral geometry as guinea-pig ventricular myocytes (100×20×8 μm3) except for its smaller scale in cell dimension (see Matsuoka et al. 2003). Given the surface area provided by the capacitance and the hexahedral geometry, the cell dimension was determined and used to calculate the cell volume. The Vi for ion diffusion was assumed to be 50% of the cell volume (Matsuoka et al. 2003). We also divided the SR into a Ca2+-release site in the junctional space and a Ca2+-uptake site in the deep cytoplasmic space (Mackenzie et al. 2001). The cell capacitance and the volume of each compartment are summarized in table 2.
(b) Ca2+-binding proteins
The amounts of Ca2+ that bound to troponin, calmodulin and calsequestrin were estimated using the approach described by Luo & Rudy (1994). We assumed that the rates of Ca2+ binding are so fast that the concentration of free calcium gets equilibrium with the Ca2+-binding proteins instantaneously. The concentrations of free calcium and Ca2+-binding proteins satisfy the following equations at equilibrium (table 3):(2.1)
(c) Calculation of the membrane potential and internal ion concentrations
(i) Membrane potential
Time-dependent changes in the membrane potential are described by the following equation:(2.6)where Iext is the current applied through the electrode by the current clamp or whole-cell voltage-clamp circuitry. Itot includes an inward rectifier K+ current (IK1), an L-type Ca2+ current (ICaL), a voltage-dependent and dihydropyridine (DHP)-resistant current (IVDDR), an autonomous inward current (IAI), a Na+/Ca2+ exchange current (INaCa), a Na+/K+ pump current (INaK) and a plasmalemmal Ca2+ pump current (IPMCA):(2.7)
(ii) Internal ion concentration
The cytosolic ionic concentrations were determined by the net ion fluxes across the plasma and SR membranes. The net ion fluxes were separated into three ionic components (Na+, K+ and Ca2+) based on the ionic selectivity of each ion channels, exchanger and pumps. The ionic component of each channel current is described by the constant field equation (see equation (2.14)).(2.8)
(d) Ion channels
(i) Inward rectifier K+ current (IK1)
The IK1 has been reported in cultured ICCs from the mouse small intestine (Koh et al. 1998). The IK1 is thought to repolarize the pacemaker potential. We adopted the Kyoto model (Matsuoka et al. 2003) and modified it to reproduce the pacemaker activity of ICCs, as described below (table 4):
State C goes into state O reversibly,State O also goes into state B reversibly,We assumed the transition between state O and B occurs instantaneously.(2.15)
(ii) L-type Ca2+ current (ICaL)
The depolarization of an ICC is thought to activate the ICaL in the neighbouring ICC (Kim et al. 2002). The presence of an L-type Ca2+ channel was clearly demonstrated by Cho & Daniel (2005) using the double-immunofluorescence labelling method. We adopted the classical two-state model and used the rate constants obtained from the voltage-clamp data of Kim et al. (2002). In addition, we added the Ca2+-dependent inactivation kinetics based on the Kyoto model (Matsuoka et al. 2003).(2.22)
(iii) Voltage-dependent and DHP-resistant current (IVDDR)
The IVDDR is thought to contribute to slow wave propagation in the GI tract. The IVDDR was fully studied and described by Kim et al. (2002) in the ICCs isolated from mouse colon and small intestine. We used the parameter values obtained from their results to reconstruct the IVDDR:(2.30)
(iv) Autonomous inward current (IAI)
Goto et al. (2004) identified a large transient inward current evoked by depolarization under voltage-clamp conditions (see figure 1). They named the current as the autonomous inward current (IAI) in that it shows an inward current with an autonomous time course by depolarizing clamp pulses. After IAI was triggered, it took a regenerative time course and lasted about 500 ms. The reversal potential was around +3 mV suggesting that IAI is a non-selective cation current. These authors proposed the IAI as the pacemaker current generating the spontaneous depolarization of ICCs without an electrical stimulus. We added a [Ca2+]i-dependent activation (see equation (2.39)) to reproduce a current with an autonomous time course.(2.35)
(e) Exchanger and pumps
(i) Na+/Ca2+ exchange current (INaCa)
The kinetic model and scheme are basically identical to those of the Kyoto model (Matsuoka et al. 2003). The six-state model of Na+/Ca2+ exchange was lumped into a two-state model according to Powell et al. (1993). The Michaelis constant (Km) and conversion factor (PNaCa) were adjusted empirically to reproduce the repetitive and stable firing of pacemaker potentials (see figure 1; table 5)
(ii) Na+/K+ pump current (INaK)
Electrogenic Na+/K+ pump extrudes three Na+ ions in exchange for two K+ ions generating a net outward current. As the kinetics of the Na+/K+ pump have not been studied well in ICCs, we employed the model of Sakai et al. (1996) obtained from rabbit sino-atrial node cells. The PNaK was adjusted to maintain the [Na+]i below 20 mM against the large Na+ influx by IAI during the depolarizing phase of a pacemaker potential. The Michaelis constant (Km) was also adjusted to fit the experimental results.(2.51)
(iii) Plasmalemmal Ca2+ pump current (IPMCA)
The plasmalemmal Ca2+ pump was clearly demonstrated in the ICCs by Cho & Daniel (2005) using the double-immunofluorescence labelling technique. However, the kinetic properties of the pump are not yet well understood. For this reason, we employed a general approach for the modelling of Ca2+ pumps (Fridlyand et al. 2003). The half-activation calcium concentration was set to 8 μM to prevent an excessive Ca2+ rise in the cytosolic space of ICCs.(2.52)
(f) IP3 receptor channel and SR Ca2+ dynamics
(i) SR Ca2+ pump current (Iup)
The Ca2+ handling by the SR Ca2+ pump has been suggested to play a key role in the regulation of Ca2+-dependent pacemaker currents, which were proposed to make a pacemaker depolarization (Hirst et al. 2002; Goto et al. 2004). The Iup was calculated according to the model of Hilgemann & Noble (1987). The basic scheme is identical to that of the Kyoto model (Matsuoka et al. 2003) with a slight modification to the Km values and conversion factor (tables 6 and 7).
(ii) Ca2+ transfer from the SR uptake site to the release site (Itr)
In the cardiac myocyte modelling, the SR Ca2+ pool was divided into the uptake and release sites to describe the underlying basis for the mechanical restitution and force–frequency relationship (Luo & Rudy 1994; Matsuoka et al. 2003). There is also evidence that a force–frequency relationship exists in the gastric smooth muscle cells, a similar cell type to ICC (Fukuta et al. 2002). Therefore, we separated the SR Ca2+ pool of ICC into two different regions. The Ca2+ transfer is suggested to move Ca2+ from the uptake site to the release site of the SR to provide Ca2+ for the next release. We used the scheme identical to that of the Kyoto model and adjusted the conversion or amplitude factor (Ptr) to fit the time course of the autonomous inward current (see figure 3) representing the cytosolic Ca2+ transient.(2.64)
(iii) Ca2+ leak from the SR (Ileak)
The Ileak was also taken from a scheme identical to that of the Kyoto model, and the conversion factor (Pleak) was adjusted to fit the time course of the cytosolic Ca2+ transient.(2.65)
(iv) IP3-mediated Ca2+ release from the SR (IIP3R)
The IP3-mediated Ca2+ release has been suggested to mediate the generation of pacemaker potential via the activation of a Ca2+-dependent inward current (Ward et al. 2000; Malysz et al. 2001; Goto et al. 2004). There is an increasing body of evidence that the IIP3R in the SR is dependent on both the Ca2+ and IP3 concentrations (Bezprozvanny et al. 1991; Iino et al. 1993; Marchant & Taylor 1997; Taylor & Laude 2002). We employed the model of Marchant & Taylor (1997), which describes the opening of IP3 receptors by sequential binding of IP3 and Ca2+. In their scheme, the binding of IP3 rapidly changes the conformation of the receptor to expose a Ca2+-binding site; Ca2+ then binds to this newly exposed site, and the channel opens allowing Ca2+ to pass through. They proposed that three or four subunits of the IP3 receptor must bind IP3 before the channel opens and releases Ca2+ into the cytosol. We added another term, Ca2+ concentration at the release site of the SR ([Ca2+]rel), to allow the rate of Ca2+ release to be dependent on the fluctuation of the SR Ca2+ pool (see equation (2.67)). All the rate constants were modified to reproduce the IAI.
(v) Ca2+ concentration in the SR
The Ca2+ concentrations of the SR uptake and release sites were calculated using the following equations:(2.71)
(g) IP3 metabolism
The IP3 plays a central role in mobilizing Ca2+ in eukaryotic cells (Berridge & Irvine 1989; Rana & Hokin 1990). The binding of hormone to the receptors on a cell surface activates the phospholipase C (PLC), which subsequently hydrolyses PIP2 in the plasma membrane into the IP3 and diacylglycerol. The IP3 finally opens a channel on the SR to release the stored Ca2+ into the cytosolic space (IP3-mediated Ca2+ release) and IP3 is then recycled to the plasma membrane after its degradation to the inactive forms. There is an increasing body of evidence that IP3 formation is also dependent on the membrane depolarization (Vergara et al. 1985; Best & Bolton 1986; Wang et al. 1995; Ganitkevich & Isenberg 1996; Goto et al. 2004). We created a kinetic scheme to reproduce the depolarization-evoked rise of IP3-mediated Ca2+ release in the ICCs from mouse small intestine (Goto et al. 2004). The following scheme simplifies the metabolic pathways of inositol phosphates (PIP2, IP3, IP4 and other metabolites):
The time-dependent changes of inositol phosphates can be described by the following equations:(2.73)(2.74)(2.75)The total amount of available inositol phosphates was assumed to have a constant value (3.3 μM) during a normal pacemaker activity. The basal level of [IP3] (less than 50 nM) calculated from our simulation was in a range similar to the value (10±3 nM) obtained by Wang et al. (1995).(2.76)The rate constants of IP3 production were set to be voltage- and Ca2+-dependent (Vergara et al. 1985; Best & Bolton 1986; Biden & Wollheim 1986; Takazawa et al. 1990; Wang et al. 1995; Ganitkevich & Isenberg 1996; Goto et al. 2004). The voltage dependence was given to the forward and backward rate constants to reproduce the voltage-dependent activation of the autonomous inward currents obtained by Goto et al. (2004). The forward rate constant was set to increase exponentially as the membrane potential goes into the positive direction, while the backward rate constant was set to decrease exponentially. The Ca2+ dependence was also given to the forward rate constant of IP3 production. Since the Hill function has been used to describe the Ca2+ dependence on the IP3 production (Biden & Wollheim 1986; Takazawa et al. 1990; Wang et al. 1995; Allen et al. 1997), the half-activation by Ca2+ (KmCai) was taken to model the IP3 production. KmCai was set to 10 nM, which is close to the basal level of [IP3] obtained by Wang et al. (1995). As for the remaining part of IP3 metabolic pathways, we set the rate constants empirically to fit the shape of experimentally obtained pacemaker potentials.(2.77)
(a) Reconstruction of pacemaker activity in the mouse ICCs
We simulated the electrical activity of ICCs from mouse small intestine with the integration of various cellular events as described in §2. As a result, spontaneous and regularly firing APs (pacemaker potential) were successfully reproduced (figure 1). The resting and overshoot potentials were around −78 and +1.2 mV, respectively. The duration of depolarizations was around 639 ms as measured at 50% repolarization. The frequency of pacemaker potentials was around 20 min−1. Compared with the experimental recordings by Goto et al. (2004), the frequency is higher in our model cell (20 versus 16.2 min−1, the model simulation versus the experimental recording, respectively) and the duration (639 versus 489.1 ms) is also longer in the model cell. The repetitive firing of APs was very stable throughout the simulation experiment, which exceeded the equivalent of 10 min on a cellular time-scale.
A single pacemaker potential from the mouse ICCs was compared with that from the model simulation (figure 2). The shape of the pacemaker potential is apparently similar between the two recordings (figure 2, left panel).
Both the recordings show a low resting potential, abrupt depolarization and repolarization, and a long plateau characteristic of ICCs (Dickens et al. 2000). The rising phase of the spontaneous depolarization was also compared between the two recordings in an expanded view (figure 2, right panel). The maximum rate of rise is 7.1 V s−1 in the experimental recording, while it is 4.32 V s−1 in the model simulation. Goto et al. (2004) reported other cell groups with a different range of the maximum rate of rise (3.6±0.7 V s−1 at around −22.2 mV, n=5) in the same study. The duration at 50% repolarization was also different for the recording in figure 2 (489.1±32.8 versus 269.5±29.7 ms, n=5).
(b) Activation of the IAI by depolarization
The different value of the maximum rate of rise among the cells indicates that the amplitude of the pacemaker inward current is also variable. Figure 3a shows the activation of inward currents in a mouse ICC by depolarization with a conventional voltage-clamp protocol. The holding potential was −80 mV. As a 30 mV depolarization with a duration of 200 ms was applied, a large inward current was evoked after a delay of approximately 20 ms. This large inward current was also evoked with smaller amplitude by larger depolarizations. Interestingly, this inward current showed an autonomous time course lasting about 500 ms, irrespective of the duration of the depolarizing pulse. The delay in the development of inward current was shortened with a larger depolarization. The time course of the inward current after repolarization to the holding potential was also not changed. Based on these characteristic features, Goto et al. (2004) called this inward current the autonomous inward current (IAI). Figure 3b shows the recordings made by the model simulation. The model simulation also shows similar inward currents in response to a series of membrane depolarizations. The time course of current relaxation was found to differ between the experimental and simulation recordings. The current size also differed between the two, with smaller amplitude in the simulation recording. The latter observation indicates that the difference in the values for the maximum rate of rise (see figure 2) between two recordings is derived from the different size of the IAI.
(c) Changes in cellular parameters during the occurrence of a pacemaker potential
Figure 4 shows the time-dependent changes in cellular parameters such as membrane currents, [Ca2+]i, [IP3], [Ca2+]rel and availability of the IP3R channel (Po,IP3R). The data demonstrate that the major conductance causing a pacemaker depolarization in ICCs is IAI and that the conductance causing the repolarization is IK1 (figure 4a). The close relationship among the time courses of membrane potential, [IP3], [Ca2+]i and Po,IP3R favours the hypothesis that cyclic changes in IP3 and Ca2+i might play a central role in the pacemaker mechanism (Van Helden et al. 2000; Goto et al. 2004).
4. Discussion and conclusions
Our model faithfully reproduces the pacemaker activity of the intestinal ICCs. Spontaneous and repetitive firing of APs continues as long as the simulation is running. The initial triggering event is thought to begin with the Ca2+ leak from the SR and the DHP-resistant pathway (IVDDR) (Kim et al. 2002). In fact, the removal of the Ca2+ leak from the SR and plasma membrane abolished the spontaneous activity of the model (data not shown). A local rise in Ca2+ concentration near the SR activates the IP3-mediated Ca2+ release (Berridge 1990; Bezprozvanny et al. 1991; Jaffe 1991; Iino et al. 1993; Marchant & Taylor 1997; Taylor & Laude 2002). In addition, cytosolic Ca2+ concentration enhances the IP3 production by the activation of IP3 3-kinase (Biden & Wollheim 1986; Takazawa et al. 1990; Wang et al. 1995). Taken together, Ca2+ release and IP3 production work in a cooperative manner and make a positive feedback to increase [Ca2+]i. An increase in the Ca2+ concentration in the subsarcolemmal region to a threshold value activates Ca2+-activated channels, which depolarize the cell membrane (Hirst et al. 2002; Goto et al. 2004). The membrane depolarization has an additional effect of enhancing IP3 production (Vergara et al. 1985; Best & Bolton 1986; Wang et al. 1995; Ganitkevich & Isenberg 1996; Goto et al. 2004). A prolonged increase of IP3 production by membrane depolarization creates the plateau phase of the pacemaker potential. The depletion of the Ca2+ pool and the gradual decrease of the available IP3 pool are thought to terminate the plateau phase. A repolarization to the threshold level finally activates the inward rectifier K+ channel, which restores the membrane potential to the resting level.
Although our model reproduces the pacemaker activity faithfully, it is unclear whether the parameter values used here appropriately reflect the real dynamics and homeostasis of a cell. There are so many interactions between parameter values that it is nearly impossible to obtain a single solution set that fits the experimental results. Each component of modelling should be validated by carefully designed experiments. For example, the effect of [Ca2+]rel on the recovery of IP3 receptor channel from the inactivated state is based on many cardiac models in which there is more direct evidence for SR Ca2+-load dependence on ryanodine release channels (Matsuda & Noma 1984). It should be validated experimentally. Even though our model needs additional validation and improvement in the near future, it is thought that the model has a value in that it provides a system to test the accumulated results and hypotheses on the pacemaker activity of ICCs.
This work was supported by funds from the Advanced Backbone IT Technology Development Project of the Ministry of Information and Communication (IMT-2000-C3-5; IMT-2000-C3-3) and the Leading Project for Biosimulation from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
One contribution of 13 to a Theme Issue ‘Biomathematical modelling I’.
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