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CALL FOR PAPERS
Special Section On Mitochondrial Modeling and Function
1Department of Bioengineering, University of California San Diego, La Jolla; 2David Geffen School of Medicine at University of California Los Angeles, California
Submitted 31 March 2006 ; accepted in final form 24 February 2007
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ABSTRACT |
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 TOP ABSTRACT GLOSSARYMETHODSRESULTSAPPENDIX AGRANTSREFERENCES |
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ATP-sensitive K+ channel; creatine and adenylate kinase reactions; phosphatidylinositol phosphates; heart; mathematical model
The epicardial, endocardial, and midmyocardial surfaces also respond differently to pharmacological agents and pathophysiological states (35, 38, 50, 51, 75, 88). The observations suggest that despite the greater susceptibility of endocardium to metabolic effects of ischemia, the electrophysiological changes evoked in epicardium are greater, and that this phenomenon may facilitate reentrant arrhythmias (35, 38, 50, 51, 72, 75). It has been demonstrated experimentally that during ATP depletion, the shortening in action potential duration is significantly greater in epicardial cells than in endocardial (35, 75). Various explanations have been offered from in vivo studies and from multicellular ventricular preparations, including effects of cavity blood (33, 54), Thebesian blood flow (30), greater capacity of the subendocardium for anaerobic metabolism (4, 40), and resistance of Purkinje fibers to the effects of hypoxia and ischemia (38). However, it is impossible to derive an understanding for the inherent regional electrophysiological cell properties from tissue preparations, which have extracellular ionic and electrotonic influences. In this contest, several reports (35, 50, 51, 75) add a further twist to the story by revealing a new role for KATP channels when ATP is depleted and these channels are activated. These reports suggest that a differential sensitivity of KATP channels to ATP among the cell subtypes might be partially responsible for the greater action potential shortening in epicardial myocytes during ischemia. To test the above hypothesis, Furukawa et al. (35) measured the effects of ATP depletion on the KATP channel activity in single cells isolated from epicardial and endocardial surfaces. They demonstrated that the reduction in intracellular ATP evoked currents through KATP channel to a greater magnitude in epicardial myocytes than in endocardial. However, the physiological basis for the site-related differential sensitivity of KATP channels to ATP remains contradictable and unclear.
The contribution of active ion transport and the physiological distinctions between ventricular epicardial, midmyocardial, and endocardial myocytes to the development of the action potential has been established in cell modeling as well (32, 45, 59, 62, 70, 73, 74, 78, 80, 87, 91). In these models the empirical functions and parameters are, in general, fitted from data obtained under normal physiological conditions when ATP is high and KATP channels are closed. However, little attention has been paid to the electrophysiological effects of anoxia (fall in [ATP]tot/[ADP]tot ratio and opening of ATP-sensitive K+ channels), acidosis (intra- and extracellular pH drop), and hyperkalemia (accumulation of extracellular K+) among the cell types, despite their obvious importance in understanding pathologies such as ischemia (10, 17, 21, 39, 63, 65, 67, 81).
In the present study we used the modeling approach to investigate the mechanisms regulating excitation-metabolic coupling in rabbit epicardial, midmyocardial, and endocardial ventricular myocytes under normal conditions and during 20 min of simulated ischemia. Here we extended the LabHEART model (74) by incorporating equations for Ca2+ and Mg2+ buffering by ATP and ADP (65) and equations describing the nucleotide regulation of several ion channels and transporters [KATP channel, L-type Ca2+ channel, Na+-K+-ATPase, sarcolemmal Ca2+-ATPase, SR Ca2+-ATPase] (20, 67). In the model, creatine and adenylate kinase (CK and AK, respectively) reactions, known to communicate the intracellular ATPases flux changes, were also included (16, 17). To simulate pHi regulation in control conditions and during ischemia, we used the approaches of Iotti et al. (43) and Shaw and Rudy (81).
In agreement with experiments of Furukawa et al. (35) and Light et al. (56), our studies revealed that in isolated membrane patches the KATP channels are activated by smallest reduction in ATP in epicardial and largest in endocardial myocytes at normal ligand (ADP, AMP, PCr, Cr), ionic (Na+, K+, Ca2+), Pi, total Mg2+, and pHi diastolic levels. Here, our analysis suggests that regional variations only in Kir6.2 half-saturation constant for ATP (kATP(4–)), but not in SUR2A half-saturation constant (kMgADP(–)), could be one of the reasons for the observed different site-related sensitivity of the channel to ATP. Thus, we concluded that in normal conditions only KATP ionophore (ATP-binding subunit) might differ among the cell types while these cells probably share common regulatory subunit (SUR2A). In agreement with experiment (41) our studies also suggest that during ischemia, the inhomogeneous accumulation of metabolites in the sarcolemma may alter in a very irregular manner the normal channel sensitivity to ATP (i.e., epicardial and endocardial kATP(4–) values) through metabolic interactions with the endogenous lipid phosphoinositide (PI) cascade. This in turn may cause differential action potential shortening among the cell subtypes. Preliminary results of this work have been presented to the Biophysical Society in abstract form (58).
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GLOSSARY |
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TOPABSTRACTGLOSSARYMETHODSRESULTSAPPENDIX AGRANTSREFERENCES |
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Volumes, Areas, and Capacity
Concentrations
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Membrane Currents, SR Fluxes, and Parameters
Dissociation, Rate, and Equilibrium Constants
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METHODS |
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TOPABSTRACTGLOSSARYMETHODSRESULTSAPPENDIX AGRANTSREFERENCES |
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Experimental data also suggest that in cardiac myocytes, ATP (as MgATP2–) and ADP (as MgADP–) drive a number of enzymes, transporters (Na+-K+-ATPase and sarcolemmal and SR Ca2+-ATPases) and channels (ATP-sensitive K+ channels, L-type Ca2+ channels) (14, 15, 43, 47, 71). In this study, to model the transporter nucleotide regulation, we modified the ATP/ADP kinetic equations of Cortassa et al. (20) for Na+-K+ pump current, sarcolemmal Ca2+ pump current, and SR Ca2+-ATPase pump current, assuming INaK, Ip(Ca), and Jup are regulated by MgATP2– and MgADP–:
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Here it is important to emphasize that model agreement with experiment will be strongly dependent on the use of realistic ligand (ATP, ADP, AMP, Pi, PCr, Cr) and extracellular and intracellular ionic (Na+, K+, Mg2+, H+) concentrations under normal conditions and during ischemia (14, 15, 43, 47, 52, 61, 77). However, experimental data for the nucleotide, PCr, Cr, Pi, and ionic concentration changes in rabbits during ischemia are limited (48, 61, 75). Thus, to be able to calculate, if experimental data were not available, the resting nucleotide, metabolite, or Mg2+ levels, we included in the model the basic reactions (ATP hydrolysis, CK and AK catalysis) known to communicate the intracellular ATPase fluxes:
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Recent studies also suggest that the exact knowledge of the apparent equilibrium constant of creatine kinase reaction (K'CK) is essential for accurate in vivo quantification of the total nucleotide, phosphate, and total Mg2+ levels (43). Therefore, in the present study, we used the model of CK reaction from Iotti et al. (43) to estimate K'CK in control conditions when diastolic [K+]i, [Na+]i, [Mg2+]i, and pHi levels are known:
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The coefficient values (a–j) describing K'CK in the pHi and pMg range 5–8 and 2–4, respectively, are shown in Table 2. Our pHi and pMg values are within that range (see Table 3, column 2). The using the accurate model for the creatine kinase reaction in normoxia was essential, because subsequently this allowed evaluating the nucleotide and free Mg2+ concentrations as well the apparent K'CK constant at 20 min of ischemia (see Table 3).
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Finally, we need to add that in this study the effects of acidosis among the cell subtypes were simulated using the approach of Shaw and Rudy (81): 1) the maximum conductance of L-type Ca2+ was reduced by 50%; and 2) the maximum conductance of INa was reduced by 25%, and its voltage-current curve was shifted to the right by 3.4 mV.
Rapid equilibrium for Ca2+ and Mg2+ buffering by ATP and ADP.
Our previous studies have demonstrated that the Eqs. 1–11 are biologically accurate but complicated computationally (65). In this article we sought to minimize the computational complexity by assuming that Ca2+ and Mg2+ buffering by ATP and ADP occur on much faster time scales than other excitation-contraction coupling processes included in the new ionic-metabolic model. Thus the following equations can be written:
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RESULTS |
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TOPABSTRACTGLOSSARYMETHODSRESULTSAPPENDIX AGRANTSREFERENCES |
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140 mM) containing 10, 25, 50, 100, 250, 500, and 1,000 µM free ATP while total intracellular Mg2+, ADP, AMP, PCr, Pi, extra- and intracellular K+, Na+, Ca2+, and pHi remained at normal diastolic levels. The data from Furukawa et al. (35) (see diamonds and triangles in Fig. 3A) suggest that the open probability of KATP channels was reduced in a dose-response fashion among the cell subtypes. Figure 3A also shows that the concentration of ATP that produced half-maximal inhibition of the channel was 23.6 µM in endocardial and 97 µM in epicardial patches. In 1999, Light et al. (56) recorded the ATP concentration-dependent relationship for KATP channel in isolated rabbit ventricle myocytes without regard to the cell location (see circles in Fig. 3B). Note that the data from Light et al. (56) show that the rabbit dose-response curve (IC50 21 µM) is quite similar to the ATP dose-response relation in cat endocardial myocytes (IC50 23.6 µM). Therefore, this was our justification for using cat data to fit the ionic-metabolic model in rabbits among the cell subtypes. Figure 3 shows our attempt to create simulations that quantitatively approximate the reported experimental data in epicardial and endocardial cat and rabbit myocytes. During this experiment, in each point on the simulated curves, the relative KATP current (IK(ATP)/IK(ATP=0)) was computed in steady state at fixed total ATP while total Mg2+ and ADP, [PCr]tot, [Cr]i, [AMP]tot, [Pi], and H+, K+, Na+, and Ca2+ diastolic concentrations were kept normal (see Table 3, column 2).
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50% that in epicardial cells (see Table 4). In addition, the results in Fig. 3A also indicate that the simulated relative currents in response to rhythmically applied pulses (1-Hz, 19–20 s) approached 1 at 0.1 µM intracellular free ATP in the three myocyte subtypes but were close to 0 at 1 mM [ATP4–]i in epicardial, at 650 µM [ATP4–]i in midmyocardial, and at 300 µM [ATP4–]i in endocardial cells.
Modeling normoxia in rabbit endocardial, midmyocardial, and epicardial ventricular myocytes.
Many experimental protocols indicate that in normal conditions the KATP channel activity is inhibited throughout the cell subtypes (see Fig. 1A) (8, 31, 35, 69, 71). To test whether our whole-cell ionic-metabolic model is able to predict that IK(ATP) current is almost inactive in healthy ventricular tissue, we calculated this current during a single beat. Graphs (see Fig. 4A) demonstrate that the channel activity in the three regions was low during the cell excitation. These results also indicate that the predicted epicardial, midmyocardial, and endocardial steady-state KATP current after 20 ms reached peaks of 0.12, 0.027, and 0.004, respectively, and that the current duration was shortest in epicardial and longest in midmyocardial myocytes. In addition, these KATP site-related currents had minimal contribution to the action potential shape when [ATP4–]i was high (390 µM) and [MgADP–]i low (67 µM) (see Fig. 4B). The channel parameter values and resting ligand and ionic concentrations used in this numerical experiment are shown in Tables 1 and 2, 4, and Table 3 (column 2).
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33 times) with [Mg2+]tot constant and the drop in pHi: 1) decreased APD90 in epicardium from 146 to 71 ms, i.e., 2 times (see solid gray and dashed black lines in Fig. 5Ba) and in endocardium from 195 to 107 ms, i.e., 1.82 times (see dotted gray and dash-dotted black lines in Fig. 5Ba); 2) did not affect the normal resting potentials; and 3) increased the steady-state outward K+ current in subepicardial cell more sensitively than in subendocardial cell (see Fig. 5B, b and c). However, it is important to stress that according to the data of Qi et al. (75), the endocardial APD90 decreased 1.32 times while the epicardial APD90 drop was much more pronounced, i.e., 2.6 times. Thus, questions arising here were the following: 1) What might be the physiological reason(s) for the observed different subendocardial and subepicardial APD90 shortening? and 2) Does our ionic-metabolic model help to explain (at least partially) this phenomenon that has puzzled people for years?
To further explore the ionic basis for the differential electrophysiological responses to ischemia at the two sites, Qi et al. (75) used specific IK(ATP) channel blockers. They demonstrated that, although the changes in steady-state outward K+ current were the sum of the changes of many K+ currents (Ito, IKp, IKr, IKs, IK1, INaK, IK(ATP)), the observed increases in the total outward K+ current among the cell types were mainly due to KATP channel opening (see Fig. 5A, b and c). However, why ischemia evoked a greater increase in the epicardial outward K+ current and a greater decrease in the epicardial APD90 remained unclear. Here, we hypothesized that ischemia may alter the normal ATP sensitivity of KATP channels among the cell subtypes in a very irregular manner, contributing to the differential response of those myocytes. To test this hypothesis we performed another set of calculations searching for new epicardial and endocardial kATP(4–), kMgADP(–), go, and gd parameter values. Surprisingly, our model predicted that in epicardial myocytes, by varying only the Kir6.2 half-maximal saturation constant (i.e., increasing kATP(4–) from 0.6 mM up to infinity), we were able to shorten the APD90 more significantly (2.43 times) (see solid black line in Fig. 5Ba). Furthermore, we searched for values of epi kATP(4–) that are reasonably close to 600 µM and that yield action potential shortening sufficiently close to the theoretical maximum. Thus, the model predicts that APD90 with kATP(4–) 8 mM is quite similar to that with kATP(4–) 
. The results also revealed that the increase in steady-state outward K+ current in subepicardial cells was significantly greater with kATP(4–) at 8 mM than predicted with kATP(4–) at 600 µM (see solid black and dashed black lines in Fig. 5Bb). Furthermore, our studies in the endocardium revealed that the normal APD90 dropped from 195 to 127 ms (1.54 times) when kATP(4–) decreased from 131 µM to zero. The simulations also showed that APD90 remained at 127 ms with kATP(4–) at 20 µM and that there was little change in APD90 with lower values of kATP(4–) (see Fig. 5Ba, dotted black line). While predicted endocardial outward K+ current was lower when calculated with the lower kATP(4–), the predicted increase in this current at 20 min ischemia was insignificant at both kATP(4–) values (131 µM or 20 µM) (see Fig. 5Bc, dotted black and dash-dotted black lines). In addition, our analysis suggests that changes in kMgADP(–), go, and gd alone (kMgADP(–) = 0.001–10, go = 0–1, gd = 0–1) do not seem to account for the experimental findings in either region (data not shown).
The predicted ionic currents and Ca2+ time courses in the myoplasm and sarcoplasmic reticulum in each of the cell subtypes during normal and ischemic action potentials (see solid and dotted lines in Fig. 5Ba) are shown in Fig. 6.
The simulations revealed that in control conditions (see gray graphs in Fig. 6, A and F–I), due to the higher K+ conductance in epicardium, the Ito, IKp, IKr, and IKs currents were enhanced, whereas the converse was true for INa current (epicardial GNa < endocardial GNa). Model results also demonstrate that under normal conditions, the differing outward K+ current densities among the tissue sublayers (see gray graphs in Fig. 5B, b and c) had a differential effect on the epicardial and endocardial ICa, INaCa, INaK, IK1, Ip(Ca), ICa,b, INa,b, [Ca2+]i, and [Ca2+]SR time courses (see gray graphs in Fig. 6, B–D and J–O). Finally, Fig. 6E shows that in normoxia the KATP channel activity was low in both regions.
At 20 min simulated ischemia, the epicardial and endocardial INa values were reduced due to the effects of acidosis (see Fig. 6A). The combined effects of acidosis and anoxia had a significant influence on ICa peak and duration throughout the cell subtypes (see Fig. 6B). Anoxia also resulted in activation of IK(ATP) current that was much more pronounced in epicardial myocytes (Fig. 6E). Our analysis also suggests that changes in [MgATP2–]i and [MgADP–]i most significantly affected the SR Ca2+-ATPase pump activities that in turn depleted the SR Ca2+ content and decreased the myoplasmic Ca2+ peaks in both cell subtypes (see Fig. 6, N and O). In addition, the ischemic changes in global [Ca2+]i concentration affected the efficiency of the Na+/Ca2+ exchanger (see Fig. 6C). Figure 6D shows that in both regions, the anoxic changes in MgATP2– and MgADP– significantly shortened INaK duration, increased the rest current, and decreased INaK peak while not significantly affecting Ip(Ca) time course (see Fig. 6K). The results in Fig. 6 also indicate that the changes in the ligand, [Na+]o, [Mg2+]i, pHi, and [Pi] levels and the drop in K'CK value (1.8 times) at 20 min of simulated ischemia had marked effects on Ito, IKp, IKr, IKs, IK1, ICa,b, and INa,b time courses throughout the cell subtypes. The model predictions for reduced SR content, decreased systolic Ca2+ peak, and activated IK(ATP) current during ischemia are in qualitative agreement with experiment (15, 44, 46).
DISCUSSION
Regional ionic-metabolic models in rabbit ventricular myocytes. In 2001, we extended the model of Winslow et al. (87) in dog ventricular myocytes to investigate how Ca2+ and Mg2+ buffering and transport by ATP and ADP regulate cardiac excitation-contraction coupling (65, 87) and how the fall in [ATP]tot/[ADP]tot ratio may affect the intracellular Ca2+, Mg2+, Na+, and K+ concentrations, the free and bound ATP and ADP diastolic and systolic levels, and the ionic currents in the myoplasm, submembrane space, and sarcoplasmic reticulum. The important limitation of this model was that the KATP current, known to be inactive in healthy ventricular tissue and increasingly outward with decreasing levels of ATP, was not included. For this reason recently, using a model approach aimed at simulating the underlying molecular nature of the KATP channel rather than taking a phenomenological approach, we formulated a new model for IK(ATP) regulation by intracellular free ATP and MgADP and incorporated this model into our whole-cell dog ionic-metabolic model (67). The updated model was able to reproduce quantitatively or qualitatively a sequence of events that corresponds well with published experimental data under normal conditions and during ischemia (2, 55, 69). However, we need to acknowledge that both models (65, 67) have limitations: 1) over a longer time period (t > 15 s) these models failed to achieve steady state; 2) the effects of acidosis on the ionic currents and concentrations were not included; 3) some unrealistic ligand ([ATP]tot, [ADP]tot) and extra- and intracellular ionic concentrations (Na+, K+, Mg2+, Ca2+) under normal conditions and for the duration of ischemia were used. To overcome the above limitations, in this article we extended the LabHEART model in rabbits (74), which is reasonably stable over a long time period, by incorporating equations for Ca2+ and Mg2+ buffering by ATP and ADP, and for the nucleotide regulation of KATP and L-type Ca2+ channels. Here, to further investigate and better understand how the changes in ATP and ADP during ischemia regulate the complex cellular dynamics, we updated the flux equations from Michailova et al. (65) for SERCA2a, Ip(Ca), and INaK ATPases. For this purpose, we modified the Cortassa et al. (20) ATP/ADP kinetic equations for these cytosolic ATP-consuming transporters, assuming dependence on MgATP2– and MgADP–. In addition, because we could not find experimental data for [CaATP2–]i, [ADP3–]i, [MgADP–]i, [CaADP–]i, [AMP]tot, [Cr]i, [phosphate], or total Mg2+ levels in normoxia, we included in the model expressions for total adenine, creatine, and phosphate, and CK and AK equilibrium reactions (16, 17), which allowed us to calculate these model parameters. To maximize the advantages of biophysically based modeling while minimizing computational complexity, we assumed also that Ca2+ and Mg2+ are buffered by ATP and ADP on much faster time scales than other excitation-contraction coupling processes. Finally, we used the approaches of Shaw and Rudy (81) and Iotti et al. (43) to simulate the pHi regulation in normoxia and during ischemia. To further test the assumptions that Michailova et al. (67) made about the kinetic mechanisms of nucleotide actions on the KATP channel subunits, we investigated the mechanisms regulating excitation-metabolic coupling in cardiac cells of epicardial, endocardial, and midmyocardial origin. Every attempt has been made to create simulations that quantitatively or qualitatively approximate published experimental data in normoxia or during 20 min of simulated ischemia in rabbit ventricular myocytes (35, 56, 75).
However, it is important to acknowledge that our new ionic-metabolic model also has limitations. The most important one is that this common-pool model is based on the assumption that the major factors affecting ionic channels and pumps (Ca2+, Na+, H+, ATP, ADP) are spatially uniform while recent studies suggest large Ca2+, Na+, H+, ATP, and ADP nonuniformity within the cell (11, 13, 14, 15, 16, 45, 47, 65, 66, 73, 76, 77, 79, 80, 84, 85, 87). This is probably true for the PIP levels also instead of assuming uniformity. However, we could not find experimental data suggesting PIP nonuniformity in the cell membrane or throughout the tissue regions. In addition, the assumption of CK in equilibrium imposes an unnecessary limitation, masking the real mechanism of metabolic regulation of respiration (77). Therefore, in our whole-cell model 1) the KATP channel is regulated by the cytosolic free ATP4– and MgADP– and not by the local link between the channel and neighboring mitochondria via the adenylate kinase energy transfer (1, 16, 79), and 2) the effects of local ligand and ionic concentration changes on Na+-K+- and Ca2+-ATPases were not studied (25, 68, 77). Other model limitations are: 1) the complex regulation of KATP channel by phosphoinositide, kinases and phosphatases is excessively simplified (89); 2) H+ binding to troponin C was not included (14, 17); 3) the normal diastolic [ATP4–]i, [CaATP2–]i, [MgATP2–]i, [ADP3–]i, [CaADP–]i, or [MgADP–]i are probably overestimated because some Mg2+, Na+, K+, and acidic forms of ATP and ADP have not been taken into account (see Ref. 43):
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Modeling KATP channel heterogeneity in excised membrane patches and normal intact cells. One problem has puzzled investigators for years: Why does the KATP channel gradually inactivate in MgATP-free solution when the integrity of the cell is disrupted by the excited patch method of voltage-clamp? Recently, a solution to the puzzle has been suggested (9, 12, 24, 26, 27, 42, 60, 82). It has been demonstrated that this phenomenon (known as channel run-down) is induced by wash out of phospholipids, including phosphatidylinositol-4,5-bisphosphates (PIP2) or phosphatidylinositol-4-phosphates (PIP) and is accompanied by a marked increase in the ATP sensitivity of the channel (12, 27, 37). Thus, it has been assumed that in vivo, MgATP2– might serve as a substrate maintaining the membrane concentrations of PIPs critical for channel activation. On the basis of these findings, a molecular/physical model describing how the channel activity might be regulated by the PIPs has been proposed (9, 22, 23, 26). This model suggests that the membrane-incorporated PIPs can bind to the positive charges in the cytoplasmic region of the channel's Kir6.2 subunits, stabilizing the open state of the channel and antagonizing the inhibitory effect of ATP. However, important questions still remain to be answered: What is the resting concentration of PIPs in the cell membrane and is it sufficient to account for the difference in the ATP sensitivity of the channel between the normal intact cell and excited patch? Why are KATP channels activated by a smaller reduction in intracellular ATP in epicardial cells than in endocardial cells in the excited patches?
In this paper, we used the model approach to address some of these questions. Our first goal was to estimate as accurately as possible the normal K'CK value, taking into account realistic intracellular ionic concentrations (Na+, K+, Mg2+, H+) reported in rabbit myocytes in control conditions (43, 74, 75). Here, we need to mention that we calculated normal K'CK as follows: 1) assuming pHi 7.1, not 7.4 (as reported in the study of Qi et al., Ref. 75), because the former value was the most widely reported (14, 15, 48); and 2) using a value of 1 mM for free normal Mg2+ because this value is reported in the experiment by Qi et al. (75) and is widely cited (1, 16, 17, 19, 80, 84, 86). Furthermore, we assumed that K'CK at [K+]i of 150 mM (see Ref. 43), which we used in our simulations, is approximately equal to K'CK at [K+]i of 145 mM, the value widely reported in normal rabbit myocytes (14, 15, 74, 80). Our studies revealed that the predicted normoxic [ATP]tot, [ADP]tot, [ATP4–]i, and [MgATP2–]i concentrations are comparable to experimentally measured values when pHi was 7.1, [Mg2+]i was 1 mM, and [K+]i was 150 mM (see Table 3, column 2).
An interesting theoretical result was that the model predictions were in a good agreement with experiment for intact epicardial, endocardial, and midmyocardial cells (15, 35, 77) despite use of kATP(4–), kMgADP–, gd, and go values estimated in isolated patches. The simulations demonstrated that KATP channel activity was inhibited in the endocardial patch at free ATP concentration of 390 µM and that the predicted endocardial KATP current was approximately zero in normal conditions. Although there was some channel activity in the epicardial and midmyocardial membrane patches in normal conditions, the KATP currents in all three populations of myocytes had minimal contributions to the normal action potential configuration when [ATP4–]i was 390 µM. In addition, our model predictions in normoxia were in good qualitative agreement with the experimental data of Qi et al. (75), suggesting shorter action potential duration and greater relative increase of the outward K+ current in the subepicardial myocytes and no differences in the epi and endo resting potentials. However, these model predictions presented a new question: how could we solve or explain the apparent dilemma that both in normal intact cells (where no alterations in normal MgATP2– and membrane PIPs levels occur) and in excited patches (where the gradual loss of PIPs has been suggested in the MgATP2– free solution), our model predictions were in good quantitative agreement with experiment (35, 75)? The answer is that in this study, in agreement with the experiment of Furukawa et al. (35) in excited patches, we estimated the KATP channel parameters (kATP(4–), kMgADP(–), gd, go) in the different cell types assuming [MgATP2–]i 
0 (4.49 mM). In addition, we concluded that our KATP model may still lack important channel structure or function details to account for the channel run-down, including how [MgATP2–]i directly regulates the KATP channel activity.
Another interesting model prediction, which was in good agreement with experimental data of Furukawa et al. (35) and Light et al. (56), was that in isolated membrane patches, KATP channels are activated by a smaller reduction in free ATP in epicardial and larger in endocardial cells at normal resting ligand (ADP, AMP, PCr, Cr), ionic (Na+, K+, Ca2+), Pi, total Mg2+, and pHi levels. Here, we found that variations only in the Kir6.2 half-saturation constant (kATP(4–)) among the isolated membrane cell patches, and not in SUR2A half-saturation constant (kMgADP(–)) or in the relative channels conductance (gd, go), are able to quantitatively fit these measurements (35, 56). Therefore, we concluded that only the KATP ionophore (i.e., Kir6.2 subunit) and not the regulatory subunit (i.e., SUR2A) differs among the cell subtypes. However, this new result yielded a new question: what could be the physiological reason(s) for the predicted regional differences in the KATP channel ionophore? Could it be differences in Kir6.2 subunit structure or some other mechanism regulating the channel function? Note that in the model, we assume that PIP levels are equal and spatially uniform and that there is not the inhomogeneous accumulation of metabolites throughout the regions in normoxia. New experiments should be suggested to test above hypotheses and the correctness of our model predictions. In addition, we need to emphasize that because the data in control conditions from Qi et al. (75) were incomplete, we did not attempt to reproduce these data quantitatively (see Table 5).
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The advantage of this model is its ability to examine and predict how the fall of [ATP]tot/[ADP]tot ratio regulates action potential development, outward K+ current, myoplasmic and SR Ca2+ transients, IK(ATP), and many other ionic currents in the different cell subtypes. To estimate as accurately as possible the initial ligand, [Pi], and [Mg2+]i concentrations at 20 min of ischemia, we searched the literature for such data measured in rabbits, and when such data could not be found, we calculated some values with the model (see Table 3, column 4). Note that Table 3 shows that predicted free Mg2+ concentration (2.24 mM) is comparable with the experimentally reported value (2–6 mM) (15). Here we simulated the effects of 20 min ischemia assuming [K+]i, [Na+]i, [Ca2+]i, [Ca2+]SR, and [Ca2+]o at normal diastolic levels. The main reasons for this were as follows: 1) in the experiment of Qi et al. (75), [Na+]o decreased by 20 mM; however, the changes in diastolic [K+]i, [Na+]i, [Ca2+]i, [Ca2+]SR, and [Ca2+]o levels were not reported; and 2) by assuming normal [K+]i while increasing [Na+]i up to 50 mM (see the fourth table in Iotti et al., Ref. 43), the predicted ischemic epicardial and endocardial action potentials were quite different in shape compared with those shown in the article by Qi et al. (75) (data not shown). To calculate K'CK, and total ADP and AMP ischemic values, we used Eqs. 15–19. In addition, we assumed K
K
because changes in the apparent KAK constant during ischemia are not reported (79).
Our study clearly demonstrates that models, including the individual components that retain details of underlying molecular processes regulating the channel kinetics, are of great importance in better understanding and explaining the electrical and contractile functions of the cell. Thus, our whole-cell model predicts that in endocardial myocytes at 20 min ischemia, the APD90 decreased 1.82 times when kATP(4–) was at the normal level of 131 µM. Note that in the experiment by Qi et al. (75), the reported decrease in the endo APD90 was 1.32 times. In addition, experiment suggests that in epicardial myocytes at 20 min ischemia, the APD90 decreased 2.6 times while the predicted APD90 drop in the epicardium was 2 times when kATP(4–) was at the normal value of 600 µM. For this reason we searched for new epicardial and endocardial kATP(4–), kMgADP(–), gd, and go parameter values. Surprisingly, the model predicted again that by varying only the Kir6.2 half-maximal saturation constant (i.e., increasing epi kATP(4–) from 600 µM to 8 mM and decreasing endocardial kATP(4–) from 131 µM to 20 µM), the degree of APD90 shortening during ischemia could more closely approximate the experiment, with epicardial APD90 shortening by 2.43 times and endocardial APD90 shortening by 1.54 times. The analysis also suggests that the changes in kMgADP(–), gd, and go alone (kMgADP(–) = 0.001–10, go = 0–1, gd = 0–1) did not account for the experimental findings in either myocyte layer at 20 min ischemia (data not shown). In addition, results revealed that the increase in outward epicardial K+ current was now significantly greater than that predicted with kATP(4–) 600 µM while endocardial outward K+ current decreased slightly when kATP(4–) was 20 µM (75). However, these new findings yielded a new question: What could be the physiological reason for the predicted dramatic changes in epicardial and endocardial kATP(4–) values during ischemia? A reasonable explanation has been suggested by Haruna et al. (41). They demonstrated experimentally that during ischemia, L-palmitoylcarnitine (a fatty acid metabolite) accumulates in the sarcolemma, deranging in a very irregular manner the membrane lipid environment, including the endogenous PI cascade (PIP2, PIP). Thus, we hypothesize here that during ischemia the inhomogeneous accumulation of the metabolites in the different tissue sublayers may alter differently the normal epicardial and endocardial ATP sensitivity of the KATP channel (i.e., kATP(4–) values) via the interactions with the membrane lipid environment (or PIPs) that in turn may cause differential transmural action potential shortening. New experiments must be performed to test the correctness of this hypothesis.
However, it is important to stress that 1) in epicardium the experiment suggest 2.6 times APD90 decrease during ischemia while the model predicted 2.43 times decrease at kATP(4–) of 8 mM; 2) in endocardium, APD90 dropped 1.54 times at kATP(4–) of 20 µM while the experiment suggests a decrease of 1.32 times; and 3) the predicted increases in the endocardial outward K+ current were insignificant at both kATP(4–) of 131 or 20 µM. Possible reasons for these differences might be that our KATP model may be incomplete or that during ischemia there are some other factors not yet included in our whole-cell model, such as intracellular acidosis (21), the activation of KATP channels by arachidonic acids (15, 49), or the regional variations in the other ionic currents during ischemia (53, 75), which might additionally affect the outward K+ current and subsequently the regional action potential configuration. Furthermore, because the data from Qi et al. (75) at 20 min of ischemia were also incomplete, in this study we did not attempt to reproduce quantitatively these data (see Table 6).
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