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NEUROREGULATION AND MOTILITY

Quantitative cellular description of gastric slow wave activity

Division of Bioengineering, National University of Singapore, Singapore

Submitted 13 November 2007 ; accepted in final form 11 February 2008

	ABSTRACT

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
Interstitial cells of Cajal (ICC) are responsible for the spontaneous and omnipresent electrical activity in the stomach. A quantitative description of the intracellular processes whose coordinated activity is believed to generate electrical slow waves has been developed and is presented here. In line with recent experimental evidence, the model describes how the interplay between the mitochondria and the endoplasmic reticulum in cycling intracellular Ca²⁺ provides the primary regulatory signal for the initiation of the slow wave. The major ion channels that have been identified as influencing slow wave activity have been modeled according to data obtained from isolated ICC. The model has been validated by comparing the simulated profile of the slow waves with experimental recordings and shows good correspondence in terms of frequency, amplitude, and shape in both control and pharmacologically altered conditions.

interstitial cells of Cajal; mathematical model; gastric electrophysiology

To date, a consensus has not been reached regarding the cellular mechanisms through which ICC generate slow waves, and several theories exist. Some investigators have identified a Cl^– conductance in ICC and suggest that ICC generate spontaneous Ca²⁺-dependent Cl^– currents that initiate slow wave activity (15, 43). The attribution of the spontaneous currents generated by ICC to a Cl^– conductance has been challenged by other authors on grounds of the pharmacological agent used to block it and the characteristics of the recorded current itself (36). Nevertheless the involvement of Cl^– currents in ICC electrophysiology has also been proposed by other investigators (14), but their presence and function remain unclear. Many experimental observations seem to identify the coordinated activity of mitochondria and endoplasmic reticulum (ER) in Ca²⁺ handling as a fundamental process for the generation of pacemaker signals (17, 20, 44). Koh et al. (21) proposed that a nonselective cationic conductance (NSCC), regulated by the local intracellular Ca²⁺ concentration, provides the electrical signal that initiates slow wave activity (36). After the generation of the primary pacemaker signal, a variety of voltage-dependent ion channels, mainly conducting Na⁺, Ca²⁺, and K⁺ ions and embedded in the plasma membrane of the ICC, are activated. The interactions between depolarizing and hyperpolarizing currents result in the observed slow wave profile.

Computational electrophysiological models provide a unique tool to succinctly describe an intricate network of interactions and allow the user to investigate the contribution of each component to the overall observed cellular behavior. Over the past few years, several modeling efforts have been directed toward the gastrointestinal tract at different spatial scales. At the tissue and organ level, electrophysiological descriptions of the stomach (33) and intestine (24) have been developed with the aim of studying slow wave propagation. These models, however, do not employ biophysically based descriptions of the cells whose activity is responsible for the slow waves. Instead, they generally employ a set of polynomial equations that can replicate the shape of a slow wave but do not allow for more in-depth investigations. Similarly, a cable model was developed recently by Edwards and Hirst (8) in which multiple ICC and SM cells were resistively coupled to replicate gastric slow wave activity. Although good correlations were achieved, the electrical descriptions of the two cell types did not attempt to describe the details of the underlying cellular processes.

When compared with other electrically active cell types, relatively few models of single cells from the gastrointestinal tract have appeared in the literature. Few models of gastrointestinal SM cells exist (31, 38), and only the more recent ones (6) are consistent with the current theory that SM cells are not able to produce slow waves themselves but need to be activated by the ICC network. To the best of our knowledge, only one mathematical description of an ICC has been developed to date (46). The authors describe the behavior of an intestinal ICC and carefully characterize most of the intracellular players and ion channels that are believed to participate in the generation and development of the slow waves. Nevertheless, they do not take into account the role played by the mitochondria in the generation of the slow waves despite the fact that several experiments have shown the importance of mitochondrial activity in slow wave generation (17, 44). Moreover, they do not include some of the ion channels whose presence has been confirmed in ICC, such as the Ca²⁺-activated (BK) K⁺ channels (5), ether-a-go-go (ERG) K⁺ channels (30). and Na⁺ channels (39). No electrophysiological model of a gastric ICC has been developed so far.

We therefore present here a mathematical description of single cell gastric ICC electrophysiology. The model details the main intracellular processes that are currently believed to provide the pacemaker signal as well as the membrane ion channels that have been identified and functionally characterized in ICC.

	MATERIALS AND METHODS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
Overview of the model. The model is based on a classical Hodgkin-Huxley approach where the cell membrane is described as an equivalent electrical circuit consisting of a capacitance connected in parallel with a number of variable conductances representing the different pathways for the movement of charged ions. The time dependence of the membrane potential is governed by

(1)

where V_m (in mV) represents the transmembrane potential, C_m (25 pF) is the cell capacitance, and I_ion (in pA) represents the sum of the ionic currents crossing the cell membrane.

Figure 1 presents a schematic overview of the model where the four main intracellular compartments [bulk cytoplasm, mitochondria, ER, and submembrane space (SS)] are displayed together with the ion channels and transport mechanisms included in the plasma membrane.

View larger version (38K): [in this window] [in a new window]   Fig. 1. Schematic view of the model. The intracellular space is divided in four compartments: cytoplasm, mitochondria, endoplasmic reticulum (ER), and submembrane space (SS). All membrane ion channels and transport mechanisms included in the model are depicted. Details of the Pacemaker Unit (PU) are given in Fig. 2.

The proportion of the intracellular volume occupied by the ER was set to 10%, similar to estimates made by other investigators on intestinal ICC (46), pancreatic β-cells (9), and cardiac myocytes (16). The SS is defined to be a subcellular compartment limited by the plasma membrane, mitochondria, and ER. It has been suggested that Ca²⁺ shuttling between mitochondria and ER within this enclosed space is responsible for the pacemaking capabilities of ICC (36). Quantification of the volume occupied by the SS is made difficult because of the lack of specific morphological information about its geometry. Sanders et al. (36) suggest that its volume should be "very small." Given a lack of quantitative information, a value 0.1% of the cell volume was chosen based on its ability to be able to yield appropriate physiological Ca²⁺ concentrations. The bulk cytoplasm, defined as the space available for ionic diffusion, is assumed to occupy 70% of the cellular volume. This value is slightly higher than the one used by Fall and Keizer (9; F&K) (50%) because of experimental observations on ICC that have shown a particularly "electron-dense" cytoplasm (22).

Pacemaker unit. The intracellular mechanisms responsible for the pacemaker activity in gastrointestinal ICC have been the object of controversy. Although many intracellular players have been implicated in the generation of the slow waves, several experimental observations appear to agree on identifying the interplay between mitochondria and ER in Ca²⁺ handling as the main process involved in initiating the electrical activity in gastrointestinal ICC (36, 42, 44). Based on these observations, comprehensive theories have recently been proposed (36, 20). Although the interested reader may find their detailed description and extensive references in these published reviews, here we briefly summarize pacemaking theory as it applies to the present model (see Fig. 2).

View larger version (35K): [in this window] [in a new window]   Fig. 2. Schematic view of the PU. The Ca2+ concentration in the SS is the result of the interplay between mitochondrial Ca2+ handling proteins (uniporter and Na+/Ca2+ exchanger), processes involving the ER [uptake through the sarco(endo)plasmic reticulum Ca2+-ATPase (SERCA) pump and inositol trisphosphate (IP3)-mediated release], and a small leakage between the SS and the cytoplasm. Red arrows represents Ca2+ fluxes, whereas blue arrows represent monovalent cationic fluxes (Na+ or K+).

A number of mathematical models describing intracellular Ca²⁺ dynamics have appeared in literature in the past few decades (29, 34, 37), each focusing on particular aspects of the ubiquitous role of Ca²⁺ ions in the cytoplasm. After an extensive search, the model developed by Fall and Keizer (9) (F&K) was chosen to represent the behavior of the intracellular Ca²⁺ handling proteins for the following reasons. First, it describes the activity of mitochondrial uniporter, mitochondrial Na⁺/Ca²⁺ exchanger, sarco(endo)plasmic reticulum Ca²⁺-ATPase pump, and IP₃-mediated Ca²⁺ release from the ER, all of which appear to be included in the pacemaker mechanism of gastric ICC (36). Second, the frequency of the Ca²⁺ cycling predicted by the F&K model is close to the typical frequency of the generation of the slow wave in the stomach (3 cycles/min) (35). Finally, the F&K model explicitly describes the dynamics of the mitochondrial membrane potential, since this has been shown to play a crucial role in the pacemaker activity of ICC in the guinea pig stomach (44). Briefly, the F&K model describes the behavior of the mitochondrial membrane potential in terms of the main ionic transport proteins on the mitochondrial membrane, including the F₀F₁ ATPase proton pump, the respiration-driven proton pump, the adenine nucleotide translocator, and the aforementioned Ca²⁺-handling proteins.

The F&K model is itself largely based on previous descriptions of intracellular dynamics (25, 26, 27), and interested readers may find details and validations of these models in the referenced publications. The kinetics of the F&K model were left unaltered with the exception of the parameters describing the volume fractions of each subspace, which were modified to include the SS.

Membrane ion channels. Each of the major membrane ionic conductances that have been experimentally described in ICC (shown in Fig. 1) were included in the I_ion term of Eq. 1. Details of these conductances are given below.

Calcium channels. At least two types of Ca²⁺ channel have been identified in ICC. One is a DHP-resistant conductance that was resolved with patch-clamp experiments on ICC isolated from the murine small intestine (19), whereas the other one has been molecularly classified as an L-type Ca²⁺ channel (5). We therefore include two equations, one for each Ca²⁺ channel conductance:

(2)

(3)

where I_VDDR and I_L-type represent the ionic currents flowing through the dihydropyridine-resistant and L-type channels, respectively. E_Ca is the Nernst potential for Ca²⁺ ions, whereas d_VDDR and f_VDDR represent activation and inactivation gating variables. Their steady-state equations have been taken directly from experimental values, and their time constants have been chosen to replicate the results from voltage-clamp experiments (19) (Table 1).

(4)

where [Ca]_i is the variation of cytoplasmic concentration (in nM) from the value in resting conditions while h_Ca (201.4 nM) and s_Ca (13.1 nM) are the half-concentration and slope factor, respectively. The time constant for this inactivation was chosen to be 2 ms (6). The values of the maximal conductances G_VDDR and G_L-type (3 and 2 nS, respectively) were chosen to reproduce the voltage-clamp experiments of Kim et al. (19).

Potassium channels. Voltage-dependent K⁺ channels encoded by the Kv1.1 gene have been identified in cultured murine ICC by immunohistochemical experiments using the double-labeling technique and functionally characterized with patch-clamp experiments on Xenopus ooctyes transfected with mRNA encoding canine Kv1.1 (12). We adopt the following equation

(5)

Here I_kv11 is the current flowing through these voltage-dependent K⁺ channels. G_kv11 (6.3 nS) is the maximal conductance, E_K is the Nernst potential for K⁺ ions, and d_kv11 and f_kv11 represent voltage-dependent gating variables whose parameters have been fitted to voltage-clamp experiments (12) (Table 1).

ERG channels have been proposed as important regulators of pacemaker activity in ICC (47). The characteristics of the ERG channels in ICC have been studied in detail in ICC by McKay et al. (30) from which the following equation was derived

(6)

where I_ERG is the current flowing through ERG channels. d_ERG is a voltage-dependent gating variable, and G_ERG (2.5 nS) is the maximal conductance.

The presence of BK channels has been confirmed in ICC via immunocytochemical techniques in the murine jejunum and lower esophageal sphincter (5). A similar conductance had been previously identified in canine colonic ICC (23), and no major difference between BK conductance in ICC and surrounding myocytes was found in terms of their voltage and intracellular Ca²⁺ dependency (10). The BK channels in ICC were therefore assumed to be not substantially different from those in gastric SM cells, and the description of Corrias and Buist (6) was adopted.

(7)

Here I_BK is the ionic current flowing through BK channels, G_BK (23 nS) represents the temperature-dependent maximal conductance, and P_BK is the Ca²⁺ and voltage-dependent open probability that is given by

(8)

where the values of K_BK (–17), Ca_set (100 µM), and h_BK (2) are taken from Carl et al. (4).

A background K⁺ conductance was also included in the model and was described as

(9)

where I_Kb represents the total background potassium current with a maximal conductance G_Kb of 0.15 nS.

Sodium channels. An SCN5A-encoded Na⁺ conductance has been found in human intestinal ICC by RT-PCR experiments (39). This current, believed to influence the rise phase of the slow waves, is tetrodotoxin resistant. The following equation was used to describe this current.

(10)

Here I_Na represents the ionic current flowing through Na⁺ channel, E_Na is the Nernst potential for Na⁺ ions, and d_Na and f_Na represent voltage-dependent gating variables whose parameters (Table 1) along with the maximal conductance G_Na (20 nS) have been chosen to replicate voltage-clamp experiments (39).

NSCC. NSCC are theorized to be the primary pacemaking conductance and allow ionic fluxes into the SS. The following equation was introduced to describe the NSCC channels

(11)

where I_NSCC is the ionic current, G_NSCC is the maximal conductance (12.15 nS), and E_NSCC is the reversal potential for NSCC channels that was calculated with the Goldman-Hodgkin-Katz equation with the ratio between Na⁺ and K⁺ permeability set to 1.056 in agreement with the notion that NSCC channels are substantially equally permeable to Na⁺ and K⁺. The value of G_NSCC was chosen to produce pacemaker potentials of sufficient size to activate the conductances on the plasma membrane responsible for the upstroke of the slow wave under normal conditions. P_openNSCC is the Ca²⁺-dependent open probability whose Ca²⁺ dependence has been briefly illustrated previously (36). Here a Hill equation was used to describe the experimental data:

(12)

where [Ca]_PU is the Ca²⁺ concentration in the pacemaker unit, K_{open NSCC} and h_{open NSCC} are the half-value and Hill coefficient, respectively, and their values (74.5 µM and –85, respectively) have been chosen to yield a physiological response with respect to the expected Ca²⁺ concentrations in the SS.

Chloride channels. The presence and functions of Cl^– in ICC has been the object of controversy (36). Zhu et al. (48) identified an inwardly rectifying whole cell current that the authors believe is carried by Cl^– ions. This dependence was described by the following equation

(13)

where I_Cl is the ionic current, E_Cl is the Nernst potential for Cl^– ions, G_Cl (10.1 nS) is the maximal conductance, and d_Cl is a gating variable that takes into account the mechanisms of intracellular regulation of Cl^– channels. Although not fully understood, some investigators have suggested that intracellular Ca²⁺ may play a role in regulating Cl^– currents in ICC (43, 48). Intracellular Ca²⁺ is already proposed as a regulatory component of Cl^– channels in SM cells in different organs with thresholds of activation >100 nM (3). Here the following Hill equation was used to model the steady-state effect of intracellular Ca²⁺ on the Cl^– current.

(14)

Here K_Cl-act represents the activation threshold and h_Cl represents the Hill coefficient. Their values (140 nM and 3, respectively) have been chosen in view of the intracellular Ca²⁺ concentrations experienced in ICC as well as taking into account activation thresholds measured in other cell types for this type of conductance (4). A time constant of 30 ms has been used to represent the kinetics of Cl^– channels opening in the calculation of d_Cl.

Ca²⁺ extrusion mechanism. At least two proteins have been identified in ICC for the purpose of Ca²⁺ extrusion from the cytoplasm, the plasmalemmal Ca²⁺ pump and the Na⁺/Ca²⁺ exchanger (5). Because quantitative data pertaining to the kinetics of these proteins was not available, a phenomenological description of these Ca²⁺ extrusion mechanisms was included to provide long-term intracellular homeostasis.

(15)

Here I_Ca-EXT is the Ca²⁺ efflux (in mM/s), I_Ca-EXT-MAX (0.0885 mM/s) and K_Ca-EXT (298 nM) are the maximal flux and the half-concentration, respectively.

	RESULTS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
Integration of Eq. 1 with the incorporation of the other model equations yields the ICC membrane potential as a function of time as is shown in Fig. 3.

View larger version (17K): [in this window] [in a new window]   Fig. 3. A: slow waves recorded from guinea pig antral interstitial cells of Cajal [ICC; adapted from Hirst and Edwards (13)]. B: simulated gastric slow waves obtained after 10 min of simulated slow wave activity from the presented model.

View larger version (7K): [in this window] [in a new window]   Fig. 4. Detail of a single simulated slow wave.

Involvement of mitochondrial activity in generation of slow waves. The involvement of mitochondria in the pacemaker mechanism in ICC has been the subject of several experiments. An addition of CGP-37157, a potent inhibitor of the mitochondrial Na⁺/Ca²⁺ exchanger, caused the inhibition of slow wave activity in the murine intestine (17). This was simulated by altering the conductance of the mitochondrial Na⁺/Ca²⁺ exchanger, and under these conditions the model also ceased to generate slow waves. Similarly, experiments have been performed to determine the involvement of the mitochondrial membrane potential on the generation of slow waves in the guinea pig (2, 11) and canine (45) stomach as well as in murine jejunum (44). In all cases, the addition of mitochondrial uncouplers such as CCCP caused a cessation of slow wave generation. Such an addition was simulated by decreasing the magnitude of the mitochondrial membrane potential. Again the model correctly showed a termination of slow waves generation under such conditions.

Involvement of the ER on slow wave activity. Malysz et al. (28) investigated the influence of IP₃-dependent Ca²⁺ release on slow wave activity in the murine small intestine. They found that blockade of IP₃-dependent Ca²⁺ release by Xestospongin C eventually abolished slow wave activity. Similar observations were made in similar tissues by Ward et al. (44). On the other hand, it was observed that stimulation of IP₃-dependent Ca²⁺ release by generation of IP₃ through a G protein-dependent pathway (triggered by stimulation of ₁-adrenoreceptors) caused an increase in the frequency of slow wave activity coupled with a suppressed plateau phase in the murine small intestine (28). The model was tested under both of these conditions by conducting two separate simulations, one with the value of the IP₃-mediated Ca²⁺ efflux from the ER set to zero (blockade) and the other with an increased concentration of IP₃. Figure 5, B and C, shows the results of simulated slow waves in such conditions compared with control conditions in Fig. 5A.

View larger version (17K): [in this window] [in a new window]   Fig. 5. A: simulated slow waves in control condition. B: simulated presence of Xestospongin C by setting to zero the Ca2+ release from the ER. C: simulated presence of 1-adrenoreceptor stimulation with consequent increase in IP3 level by 70 nM.

2-Aminoethoxydiphenyl borate (2-APB) is known to be a membrane-permeable agent capable of compromising Ca²⁺ handling by the ER. Recent reports suggest that it directly inhibits Ca²⁺ entry pathways in the ER (3, 7). Fukuta et al. (11) observed that addition of 2-APB caused a reversible blockade of slow wave activity in the guinea pig stomach (Fig. 6A). We simulated the presence of 2-APB by inhibiting the Ca²⁺ entry pathway in the ER. The results of these simulations showed a reversible cessation of slow wave activity (Fig. 6B) in agreement with the experimental data.

View larger version (40K): [in this window] [in a new window]   Fig. 6. A: effect of addition of 2-aminoethoxydiphenyl borate (2-APB) on ICC gastric slow waves from guinea pig adapted from Fukuta et al. (11). B: simulated slow waves in presence of 2-APB, a blocker of Ca2+ entry in ER.

	DISCUSSION

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

There exist other theories that differ from the NSCC hypothesis. Of particular note is the hypothesis that it is a Cl^– conductance and not a NSCC that is the primary pacemaker conductance (48). Should the NSCC theory be displaced in the future, this model, because of its modular nature, can easily be updated by replacing the current primary pacemaker conductance (NSCC) with an equation describing the kinetics of the alternate mechanism.

Despite this ongoing controversy about the nature of the primary pacemaker conductance, there seem to be agreement on the role played by Ca²⁺ and its handling by the ER and mitochondria in regulating ion channels and, consequently, in the generation of slow waves. The model was therefore developed starting with a validated model of mitochondrial and ER activity (9). To this, descriptions of the main membrane ion channels that have been identified in ICC were coupled. Where possible, data were adopted from isolated gastric ICC. Where direct measurements of currents from gastric ICC were not available, data from intestinal ICC or from gastric SMC were used. The equations that were chosen to describe the ion channels were validated by simulated voltage-clamp experiments, and the results were compared with published experimental data under the same conditions wherever possible.

Some debate also exists with regard to the ionic conductances that are involved in the plateau phase. In particular, the form of the depolarizing current that counterbalances the activation of K⁺ conductances during the plateau has not been resolved. It has been suggested that more than one conductance may participate and that L-type Ca²⁺ channels may have a role because of their incomplete inactivation during voltage-clamp experiments (32). The same investigators deemed it unlikely that L-type Ca²⁺ channels were the sole depolarizing factor during the plateau phase and observed that manipulation of Cl^– and Na⁺ concentrations led to modifications of the slow wave plateau. The involvement of Cl^– (20) and possibly other Na⁺-dependent membrane ionic mechanisms was therefore hypothesized. A Cl^– conductance was included in the model description because of reports of its presence and influence in ICC activity (15, 48), in particular in the regenerative component of the slow wave (14). Here the Cl^– conductance does not constitute a primary pacemaker conductance but contributes to the plateau phase of the slow wave. Other investigators have suggested that the plateau of the slow wave might arise from the summation of many pacemaker potentials entrained successively by an initial event (36). The velocity of slow wave conduction would, however, suggest that in this case either not all of the pacemaker units are activated by the initial upstroke of the slow wave as it moves through the tissue or that some pacemaker units recover quickly enough to be discharged a second time during the plateau phase. As is the case in other cell types, here the SS represents the summation of individual subcellular spaces, but it is clear that in the future further investigation is needed in this area.

The four K⁺ conductances included in the model represent those that have been well quantified through patch-clamp experiments and are all believed to provide significant contributions to the slow wave. It should be noted, however, that this is not an exhaustive list of those K⁺ channels that have been found in ICC. In particular, ATP-dependent K⁺ channels and intermediate and small Ca²⁺-activated K⁺ channels may also be included in the model description once comprehensive quantitative characterizations become available.

The model can be thought of as two tightly coupled components, the internal calcium cycling mechanisms that initiate pacemaker activity and the components on the plasma membrane that form the slow wave in response to this activity. As demonstrated in Fig. 5C, spontaneous rhythmicity can be maintained over a range of pacing frequencies by an appropriate manipulation of the appropriate calcium cycling parameters. The shape of the resulting slow wave is primarily determined by the parameters of the plasma membrane components, and, in general, these may be adjusted to alter the shape of the slow wave without a cessation of activity.

In conclusion, a quantitative model of slow wave generation by a gastric ICC has been developed. Included are descriptions of the main ion channels and intracellular processes that are believed to contribute to the generation of the slow waves. Such models represent a situation whereby the underlying biology is adequately represented and the user has complete control over all of the cellular parameters. This control allows predictive investigations to be performed in relation to determining the likely cause of a certain pathology, or the likely consequences of a certain intervention or mutation. An example of this is whether an ion channel (genetic) mutation could result in clinically relevant anomalies. Experimentally, the behavior of a channel is usually studied in isolation, whereas a model can allow a determination of how the other wild-type channels react to a given situation and in addition can predict changes in whole cell behavior.

The development of this model is seen as an initial step, and further efforts are needed to establish computational models as an important tool for investigating gastric pathophysiology. In particular, the inclusion of existing single cell models of ICC and SM cells (6) in multicellular tissue-level models may allow important insights into the spatial propagation of slow waves in health and disease. A sample implementation of the model is available from the authors upon request.

	FOOTNOTES

 

Address for reprint requests and other correspondence: M. L. Buist, Division of Bioengineering, National Univ. of Singapore, 9 Engineering Dr. 1, Singapore 117576 (e-mail: biebml{at}nus.edu.sg)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

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