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Modeling Regulation of Cardiac K_ATP and L-type Ca²⁺ Currents by ATP, ADP, and Mg²⁺

Department of Bioengineering, University of California San Diego, La Jolla, California

Correspondence: Address reprint requests to Dr. Anushka Michailova, Dept. of Bioengineering, University of California San Diego, La Jolla, CA 92093-0412. E-mail: amihaylo{at}bioeng.ucsd.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS DISCUSSION CONCLUSIONS APPENDIX I: MICHAILOVA-McCULLOCH... APPENDIX II: GLOSSARY ACKNOWLEDGEMENTS REFERENCES

 
Changes in cytosolic free Mg²⁺ and adenosine nucleotide phosphates affect cardiac excitability and contractility. To investigate how modulation by Mg²⁺, ATP, and ADP of K_ATP and L-type Ca²⁺ channels influences excitation-contraction coupling, we incorporated equations for intracellular ATP and MgADP regulation of the K_ATP current and MgATP regulation of the L-type Ca²⁺ current in an ionic-metabolic model of the canine ventricular myocyte. The new model: 1), quantitatively reproduces a dose-response relationship for the effects of changes in ATP on K_ATP current, 2), simulates effects of ADP in modulating ATP sensitivity of K_ATP channel, 3), predicts activation of Ca²⁺ current during rapid increase in MgATP, and 4), demonstrates that decreased ATP/ADP ratio with normal total Mg²⁺ or increased free Mg²⁺ with normal ATP and ADP activate K_ATP current, shorten action potential, and alter ionic currents and intracellular Ca²⁺ signals. The model predictions are in agreement with experimental data measured under normal and a variety of pathological conditions.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS DISCUSSION CONCLUSIONS APPENDIX I: MICHAILOVA-McCULLOCH... APPENDIX II: GLOSSARY ACKNOWLEDGEMENTS REFERENCES

 
Pathological changes of intracellular free and bound Mg²⁺, ATP, and ADP concentrations occur during ischemia, and reperfusion and the concentrations of these metabolites have been shown to affect the availability and activity of ATP-sensitive K⁺ and L-type Ca²⁺ channels and consequently cell excitability and contractility (Noma, 1983; Agus et al., 1989; Carmeliet, 1999; Bers, 2001; Michailova and McCulloch, 2001).

The ATP-sensitive K⁺ channels were first discovered by Noma (1983) in the plasma sarcolemmal membrane of cardiac myocytes. Normally, K_ATP channel activity is inhibited by intracellular free ATP. When ATP is depleted, sarcolemmal K_ATP channels open to hyperpolarize the cell. Thus, K_ATP channels couple cell metabolism to its electrical activity. These channels are also regulated by a variety of other intracellular and extracellular factors, including MgADP, MgATP, pH, G-proteins, adenosine, and extracellular ATP (Ashcroft and Ashcroft, 1990; Nichols and Lederer, 1991). Experimental data suggest that the sarcolemmal K_ATP channel consists of two types of subunits—pore-forming K⁺-channel subunits (Kir6) and sulphonylurea receptor subunits (SUR) (Inagaki et al., 1995). During the past few years, the cloning of these subunits has led to significant advances in our understanding of K_ATP channel structure-function relations. Both subunits are required to form a functional K_ATP channel, coassembling in an obligate 4:4 stoichiometry to form an octameric channel (Clement et al., 1997; Shyng and Nichols, 1997). Two different Kir6 subunit genes have been described, Kir6.1 and Kir6.2 (Inagaki et al., 1995; Sakura et al., 1995). Two closely related genes encoding the sulphonylurea receptors, SUR1 and SUR2, have been also cloned (Aguilar-Bryan et al., 1995; Inagaki et al., 1996). The various Kir and SUR subunits "mix and match" to form K_ATP channels with different pharmacological and nucleotide sensitivities. Comparison of the properties of cloned and wild-type K_ATP channels suggests that cardiac K_ATP channel is composed from Kir6.2 and SUR2A (Inagaki et al., 1995, 1996; Sakura et al., 1995; Aguilar-Bryan et al., 1995). A key question has been which properties of the K_ATP channel are intrinsic to Kir and which are endowed by SUR. The primary site at which intracellular free ATP acts to cause K_ATP channel inhibition appears to lie on Kir6.2 (Nichols and Lederer, 1991; Ashcroft and Gribble, 1998; Ribalet et al., 2003), whereas SUR2A is the principal target for pharmacological agents (Aguilar-Bryan et al., 1995; Inagaki et al., 1996). SUR2A also mediates the stimulatory effects of intracellular MgADP and enhances channel open probability (Nichols et al., 1996c; Tucker et al., 1997; Trapp et al., 1997; Proks and Ashcroft, 1997).

Several attempts have been made to integrate a simple Hill-type formulation for the ATP regulation of K_ATP channel current () into existing electrophysiological cell models, assuming channel availability as a function of absolute ATP concentration (), (Nichols et al., 1991a; Shaw and Rudy, 1994, 1997; Ferrero et al., 1996; Rodriguez et al., 2002; Matsuoka et al., 2003; Fridlyand et al., 2003). This was mainly because these ionic models did not calculate intracellular free ATP, MgADP, and MgADP, which are known to regulate the ligand-gated channel activity.

Noma and Shibasaki (1985) first demonstrated the ATP dependence of the L-type Ca²⁺ current, which was independent of membrane potential and protein phosphorylation. Later, using flash photolysis of caged Mg²⁺ or ATP, it was found that the rapid changes in intracellular MgATP, rather than free ATP, may increase the whole-cell current without significantly affecting gating kinetics (O'Rourke et al., 1992). The rate of increase in whole-cell current (O'Rourke et al., 1992) and probability distributions from inside-out patches (Yazawa et al., 1997) suggested that the MgATP regulation of is the result of an increase in channel availability (). To simulate the effects of changes in intracellular ATP level on L-type Ca²⁺ current, Shaw and Rudy (1997) and Matsuoka et al. (2003) used a Hill-type approximation, assuming

Therefore, the goals of this study were: 1), to formulate a new model for cardiac K_ATP current regulation by intracellular free ATP and MgADP that takes into account the octameric channel stoichiometry; 2), to formulate a new reduced-order model of the L-type Ca²⁺ channel and to incorporate the direct regulation of the L-type Ca²⁺ current by MgATP; 3), to integrate these models into our whole cell ventricular model (Michailova and McCulloch, 2001; Michailova et al., 2004b); and 4), to investigate how changes in intracellular free Mg²⁺ and adenine nucleotide phosphate levels modulate K_ATP current, Ca²⁺ current, and the integrated process of excitation-contraction coupling.

The new model reproduces experimental data on the ATP dependence of K_ATP channel activity in the presence of normal ADP and Mg²⁺ and the effects of stimulated K_ATP current on cell excitability and contractility (Nichols et al., 1991a). The model is also able to simulate and predict: a), the effects of ADP in modulating ATP sensitivity of the K_ATP channel, b), the activation of L-type Ca²⁺ current during rapid increase in intracellular MgATP concentration, and c), the effects of ATP in the absence of MgADP or of MgADP in the absence of ATP on macroscopic K_ATP current during a single beat. A reduction in total ATP/ADP ratio with normal or an increase of free Mg²⁺ with total ATP and ADP at normal levels increased shortened action potential duration, and affected ionic currents and intracellular Ca²⁺ levels. Variations in absolute ATP and ADP levels with total Mg²⁺ and ATP/ADP ratio unchanged affected K_ATP and L-type channel activity and cardiac EC coupling. Preliminary results of this work have been presented to the Biophysical Society in abstract form (Michailova et al., 2004a).

	MATHEMATICAL MODEL

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS DISCUSSION CONCLUSIONS APPENDIX I: MICHAILOVA-McCULLOCH... APPENDIX II: GLOSSARY ACKNOWLEDGEMENTS REFERENCES

 
Ionic-metabolic model
The whole-cell ionic-metabolic model is described in Michailova and McCulloch (2001). In that article, we extended the ionic model of the ventricular myocyte by Winslow et al. (1999) to examine the role of ATP and ADP as Ca²⁺ and Mg²⁺ buffers, transporters, and ion current regulators. The model cell has three spaces: subspace, myoplasm, and sarcoplasmic reticulum (see Fig. 1). Adenine nucleotides (ATP, ADP) and Mg²⁺ react and diffuse within subspace and myoplasm and not in the sarcoplasmic reticulum. Because MgATP is the preferred substrate for a large number of intracellular enzymes, ATP-dependent transporters and channels (Carmeliet, 1999; Bers, 2001), we also assumed that myoplasmic MgATP () regulates SR and sarcolemmal Ca²⁺-ATPases (see Fig. 1). The equations describing Ca²⁺ and Mg²⁺ buffering and transport by ATP and ADP in the subspace and myoplasm and the modified Winslow et al. (1999) equations for SERCA2a pump flux, current, and free subspace and myoplasmic Ca²⁺ concentrations are shown in Appendix I.

View larger version (66K): [in this window] [in a new window]   FIGURE 1  Schematic diagram illustrating Ca2+ and Mg2+ buffering and transport by ATP and ADP, adenine nucleotides regulation of ionic channels and pump, and electrophysiology in ventricular myocyte. See Appendix II for the notations of the parameters used throughout the study.

Modeling K_ATP channel availability
Taking into account experimental observations (Clement et al., 1997; Shyng and Nichols, 1997), we developed a model for the octameric cardiac K_ATP channel that contains four pore-forming subunits (Kir6.2) and four regulatory subunits (SUR2A) (Fig. 2 A). Binding of ATP to Kir6.2 and of MgADP to SUR2A is treated as instantaneous and the K_ATP channels were set on the plasma membrane (see Fig. 1). The general equation describing current density was as described in Shaw and Rudy (1997):

(1)

(2)

View larger version (35K): [in this window] [in a new window]   FIGURE 2  Schematic representation of Kir6.2 and SUR2A subunit stoichiometry in cardiac octameric KATP channel (A). It is assumed that the single ATP molecule (•) is sufficient to close the channel and that the binding of 2MgADP molecules () to one SUR2A subunit increases the channel open probability. Panel B shows and action-potential time courses (9–10 s; 1-Hz frequency) generated with original ionic-metabolic model (dashed lines) and with reduced-order model of the L-type Ca2+ channel (solid lines). Normal condition (1) is 5 mM 200 µM 0.5 mM ; metabolic inhibition (2) is 3 mM 3 mM 0.68 mM

(3)

Experimental studies also suggest that channel activation by ADP requires the presence of Mg²⁺ (Nichols et al., 1996c; Gribble et al., 1997; Ueda et al., 1997; Ashcroft and Gribble, 1998). The simplest explanation for this could be that MgADP interacts with both SUR2A nucleotide binding domains (NBDs), that is, one MgADP molecule binds to each NBD, so that two molecules bind to each SUR2A subunit. It is not known whether the interaction of one or two MgADP molecules with only one of the four SUR2A subunits is sufficient to cause channel activation (Ashcroft and Gribble, 1998). Here we assume (see Fig. 2 A) that the simultaneous binding of two MgADP molecules to one SUR2A subunit is required to increase the channel open probability, i.e.:

(4)

Therefore, the fraction of sarcolemmal K_ATP channels that are closed because there are not two MgADP sites occupied by MgADP in one SUR2A subunit is:

(5)

Because experimental studies suggest that there are four SUR2A subunits that bind MgADP independently and with equal affinity, the fraction of channels that are open because there are two MgADP molecules bound to only one SUR2A subunit is:

(6)

In this model, according to Hopkins et al. (1992), we also assume that when the channel has either no ATP or two MgADP molecules bound or two molecules MgADP bound, the channel is open with relative conductance of and respectively (Fig. 2 A). This leads to the following expression for the dependence of the aggregate channel availability () on adenine nucleotide concentrations when the two populations of K_ATP channels are open:

(7)

or

Finally, because many experimental studies suggest that a wide variety of intracellular and extracellular factors (MgATP, MgGDP, pH, G-proteins, phospholipids, extracellular ATP, adenosine) could additionally enhance channel open probability, here we also assumed that () is the relative conductance accounting for this more complex ligand-gated channel regulation.

MgATP regulation in a reduced-order model of the L-type Ca²⁺ channel
A structurally motivated Markov model of the L-type Ca²⁺ channel parameterized with single-channel patch-clamp data (Jafri et al., 1998) has been used in a number of previous modeling studies (Rice et al., 1999; Winslow et al., 1999), including our past description of metabolism (Michailova and McCulloch, 2001). Here we sought to formulate a simplified model of the L-type Ca²⁺ channel, which retains the properties of the more detailed Markov model yet reduces the complexity. To formulate the reduced-order model, we began by assuming the structure and kinetics of the more detailed Markov model (Jafri et al., 1998). From this starting point, several assumptions were made: 1), the four channel subunits gate independently of one another; 2), voltage-independent activation gating occurs independently of the voltage-dependent conformational changes in the subunits; 3), voltage-dependent inactivation occurs independently of activation and Ca²⁺-dependent inactivation; 4), transitions between normal and Ca²⁺-inactivated modes occur much more slowly than voltage-dependent gating within a mode. Assumptions 1 and 3 were made explicitly in the original Markov model (Jafri et al., 1998). Assumptions 2 and 4 are supported both by timescale differences in the rate constants of the Markov model (Jafri et al., 1998) and the single-channel data on which the Markov model was based (Imredy and Yue, 1994). From these assumptions, a simplified set of equations can be derived with gating variables representing conditional probabilities for the channel state:

(8)

(9)

(10)

(11)

(12)

where

All parameters in Eqs. 8–12 remain identical to those used previously (Winslow et al., 1999; Michailova and McCulloch, 2001). Thus, using Eqs. 8–12 and the above assumptions, the probability of gating channel becomes:

(13)

Further, taking into account experimental data (O'Rourke et al., 1992; Yazawa et al., 1997) and our whole-cell model topology (see Fig. 1) we modeled the relative Ca²⁺ channel availability () as a function of [MgATP]_ss using the Hill-type equation.

(14)

where (from Shaw and Rudy, 1997).

Thus, the effect of MgATP regulation of the L-type Ca²⁺ channel current () and K⁺ current through the channel () can be described by:

(15)

(16)

where

Validation studies demonstrated that replacing the detailed Jafri et al. (1998) model with our reduced-order model of did change the normal time course of the original channel open probability, respectively, and (Winslow et al., 1999; Michailova and McCulloch, 2001). To fix this problem we increased the original channel permeability ( and ) 1.15-fold. Our calculations (Fig. 2 B) show that now the normal time course of subspace Ca²⁺ signal is quite similar in shape, the calculated normal L-type Ca²⁺ current and action potential are only slightly modified, and or action-potential time courses in metabolically inhibited conditions are not affected. Because all subsequent simulations yielded very similar results with either approximation, only the simulations using the new reduced-order model are shown in the Results.

	RESULTS

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS DISCUSSION CONCLUSIONS APPENDIX I: MICHAILOVA-McCULLOCH... APPENDIX II: GLOSSARY ACKNOWLEDGEMENTS REFERENCES

 
Effects of ATP on I_K(ATP) and I_Ca currents, contractile activity, and action potential
The first set of the modeling results describes our attempt to create a simulation that quantitatively approximates the reported experimental data for the dependence of K_ATP channel activity in the presence of ADP and Mg²⁺ in guinea pig ventricular myocytes (Nichols et al., 1991a) because we could not find experimental data in canine ventricular myocytes. In these experiments: a), normal and were 5 mM, 200 µM, and 0.5 mM; and b), free ATP was omitted from the dialyzing solution, batting the intracellular surface of isolated myocyte, and in this way was decreased while total intracellular Mg²⁺ and ADP remained unchanged. Fig. 3 A shows that the new whole-cell model quantitatively reproduces the experimental data of Nichols et al. (1991a). The calculated relative current () in response to rhythmically applied pulses (1-Hz; 9–10 s) approaches one at 0.01 µM intracellular free ATP (or 0.5 mM ) and is close to zero at 1 mM (or 5.5 mM ). Under these conditions, the free model parameters () were estimated to be 600 µM, 400 µM, and 0.05 mS/µF. The relative conductance (0.08) and (0.89) were taken from Hopkins et al. (1992).

View larger version (33K): [in this window] [in a new window]   FIGURE 3  Effects of ATP on cardiac EC coupling. (A) Dose-response relationship for the effects of free ATP (or total ATP from 0 to 10 mM) on relative KATP channel current (). The solid line was fitted to experimental data by Nichols et al. (1991a) () with = 600 µM, = 400 µM, = 0.05 mS/µF, = 0.08, = 0.89. (B–F) Time courses of and action potential. Letters (i–v) correspond to 5, 4, 3, 2, and 0.5 mM respectively. Insets show experimental recordings (Fig. 4 B in Nichols et al., 1991a) of superimposed (panel B), twitches (panel E), and action potentials (panel F) for the effects of injection of stimulated KATP current. Letters (i–v) in insets correspond to digitized experimental plots (i, ii, vii, x, xi). Simulations are generated in response to 1-Hz pulse and model outputs at the tenth stimulus are shown only. 200 µM, 4.84 mM, 4 mM, 138 mM, 2 mM.

View larger version (29K): [in this window] [in a new window]   FIGURE 4  Dependence of KATP channel activity on ADP in the presence of Mg2+. Calculated ATP-sensitive K+ currents (5–10 s) in response to 1-Hz stimuli are shown. (A) Top, middle, and bottom traces show the shift in the sensitivity of KATP channels to ATP caused by increases in free ADP. (B) Effects of 1 mM increase or decrease in total ATP from 5 mM in the absence of ADP on KATP current. (C) Effects of 100 µM increase or decrease in total ADP from 200 µM in the absence of ATP on KATP current. 4.84 mM, 4 mM, 138 mM, 2 mM.

Modulation of ATP sensitivity of K_ATP channel by ADP in the presence of Mg²⁺
Experimental studies in rat and guinea pig ventricular myocytes (Lederer and Nichols, 1989; Weiss et al., 1992) suggest that cardiac K_ATP channels are regulated by ATP, and that this regulation is sensitive to other intracellular nucleotides, Mg²⁺ and pH. Measuring the ATP dependence of channel activity at different ADP with total ATP and Mg²⁺ constant, Lederer and Nichols (1989) observed that low concentrations of free ADP ( < 0.5 mM) increase K_ATP channel activity whereas free ADP levels higher than 0.5 mM inhibit channel activity. To test whether the ionic-metabolic model is able to reproduce these experimental data, we calculated current (5–10 s) in response to 1-Hz periodic pulse by increasing free ADP but keeping total ATP 5 mM (Fig. 4 A, top traces), 4.9 mM (Fig. 4 A, middle traces), and 4.5 mM (Fig. 4 A, bottom traces). During these simulations, free ATP levels () were 697, 741, and 786 µM (Fig. 4 A and Table 1) and total Mg²⁺ was constant at normal level (4.84 mM). Table 1 also shows estimated and The results suggest that: 1), the increase in free ADP, with < 0.5 mM and total ATP 5 or 4.9 mM, activated ; 2), the increase in free ADP, with > 0.5 mM and total ATP 4.5 mM, inhibited ; and 3), the decrease in total intracellular ATP level (from 5 to 4.5 mM) enhanced current for all free ADP concentrations.

The model was also used to simulate how alterations in intracellular ADP or Mg²⁺ in the absence of Kir6.2 inhibition ( = 0 mM) regulate current during a single beat. The simulations indicate that an increase of from 100 to 300 µM (and of from 88 to 262 µM) with intracellular unchanged (4.84 mM) increased peak current but these changes did not influence current duration (Fig. 4 C). These results also suggested that during current activation, dropped from 4.7 to 4.6 mM. In addition, the model predicted that a 50% increase or decrease of total Mg²⁺ from 4.84 mM with 200 µM has a negligible effect on current (not shown). The calculated levels here were 183, 175, and 157 µM.

Effects of total ATP and ADP on cardiac EC coupling
To test the hypothesis that absolute levels of adenine nucleotides ( and ) regulate cardiac EC coupling independently of the ratio, we performed another set of calculations, changing total ATP and ADP but keeping (25) and total Mg²⁺ (4.84 mM) constant. The outputs of the model (10th cycle; 9–10 s) in response to rhythmically applied pulses are shown in Fig. 5. The results showed that a 50% decrease (dashed-dotted lines) of from 5 mM and from 200 µM were able to affect noticeably the L-type Ca²⁺ current duration, diastolic and systolic current, and action potential shape whereas a 50% increase (dashed lines) enhanced, but not too sensitively, systolic Ca²⁺ peak, prolonged and action potential, and totally inhibited current. The predicted and during this experiment are shown in Table 2. Simulations also showed that the changes in total adenine nucleotide concentrations could markedly influence SR and subspace Ca²⁺ transients and all Ca²⁺-dependent currents— (not shown). Intracellular Na⁺ and K⁺ concentrations and were also influenced to some extent, but not greatly (not shown).

View larger version (20K): [in this window] [in a new window]   FIGURE 5  Absolute levels of ATP and ADP regulate cardiac EC coupling independently of ATP/ADP ratio. Panels A–D show effects of 50% simultaneous increase or decrease in total ATP and total ADP with unchanged on and action potential (9–10 s) in response to 1-Hz periodic pulse. (Dashed line) 7.5 mM, 300 µM. (Solid line) 5 mM, 200 µM. (Dashed-dotted line) 2.5 mM, 100 µM. Total Mg2+ 4.84 mM, ATP/ADP = 25.

View larger version (27K): [in this window] [in a new window]   FIGURE 6  Effects of cytosolic Mg2+ on cardiac EC coupling. Panels A–D show effects of changes in cytosolic Mg2+ on and action potential (9–10 s) in response to 1-Hz periodic pulse. (Dashed-dotted line) 0.2 mM, total Mg2+ 3.73 mM. (Solid line) 0.5 mM, total Mg2+ 4.84 mM. (Dashed line) 1.8 mM, total Mg2+ 6.7 mM. Total ATP 5 mM, total ADP 200 µM. Bottom trace in panel A shows an expanded view of the peak of L-type Ca2+ current. Inset in panel D is from Agus et al. (1989) and shows the suppressive effect of elevated Mg2+ levels on the experimentally recorded action potentials. The first, second, and last experimental plots (right to left) are digitized.

View larger version (23K): [in this window] [in a new window]   FIGURE 7  Panels A–H show model outputs in response to 1-Hz periodic pulse (9–10 s). Bottom trace in panel E shows an expanded view of the peak of L-type Ca2+ current. Normal conditions are 6.8 mM, 15 µM, 2 mM (solid line). Ischemia (40 s) is 5.4 mM, 30 µM, 3.165 mM (dashed line). Ischemia (10 min) is 4.6 mM, 99 µM, 3.55 mM (dashed-dotted line). Total Mg2+ 8.56 mM.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS DISCUSSION CONCLUSIONS APPENDIX I: MICHAILOVA-McCULLOCH... APPENDIX II: GLOSSARY ACKNOWLEDGEMENTS REFERENCES

We need to acknowledge here that ours is not the only model of ATP and MgADP regulation of the K_ATP channel availability. Hopkins et al. (1992) proposed a model of channel-nucleotide interaction with two kinds of ADP binding sites, regulating sarcolemmal K_ATP channel in mouse pancreatic ß-cells. In that model, one site specifically binds two MgADP molecules and increases channel opening. The other site binds either one molecule ATP or ADP and decreases channel opening. Recently, Ashcroft and Gribble (1998) showed also that the four-site ATP model was able to fit better their ATP dose-response experimental data than the one-site ATP model. Therefore, there were several reasons why the new formulation for the dependence of aggregate channel availability on nucleotide concentrations () was necessary: 1), the K_ATP channel is the octameric structure containing four Kir6.2 and four SUR2A subunits, not assumed in the Hopkins et al. (1992) model; 2), Kir6.2 subunit is the primary site for ATP binding only and not for ADP; and 3), MgADP probably interacts with both SUR2A NBDs, so that two molecules MgADP bind to each SUR2A subunit.

Furthermore, we need to stress that the gating kinetics of the single K_ATP channel is not included into our ionic-metabolic model yet. Experimental and theoretical studies (Spruce et al., 1987; Gillis et al., 1989; Davies, 1990; Davies et al., 1991, 1992; Nichols et al., 1991b; Alekseev et al., 1997; Trapp et al., 1997; Karschin et al., 1998) suggest that the kinetic behavior of K_ATP channel is complex, i.e., that the first ATP molecule is assumed to close the channel but subsequent ATP binding might then stabilize blocked channel or that MgADP binding to SUR2A subunit may affect the binding or action of ATP on Kir6.2 ATP site. A kinetic model of the ATP-dependent regulation of channel activity, based on the assumption of four sequential ATP-binding states, was suggested by Nichols et al. (1991b) for rat ventricular myocytes. The model assumes one ATP-independent open state, one ATP-dependent open state, one ATP-independent closed state as well as four ATP-dependent closed states reflecting the sequential binding of four ATP molecules to the channel. The binding of the first ATP molecule is assumed to close the channel and subsequent ATP binding might then stabilize the blocked channel.

Ca²⁺ channel model
To formulate the reduced-order Ca²⁺ channel gating model, we used the detailed Markov model (Jafri et al., 1998). We found that the channel subunit interactions influence sensitively channel open probability. For this reason we fit the Winslow et al. (1999) whole-cell Ca²⁺ current by increasing channel permeability for Ca²⁺ and K⁺ ions 1.15-fold. We concluded that the observed differences during current inactivation in normal conditions were probably due to limitations of assumptions: i), voltage-independent activation gating occurs independently of the voltage-dependent conformational changes in the subunits; ii), transitions between normal and Ca²⁺-inactivated modes occur much more slowly than voltage-dependent gating within a mode. Furthermore, in agreement with experimental data (O'Rourke et al., 1992; Yazawa et al., 1997)—MgATP regulates directly cardiac Ca²⁺ channel through a phosphorylation-independent mechanism—we assumed the channel availability () depending on subspace MgATP concentration. Our studies revealed that both approaches yield similar results under a variety of pathological conditions.

Cardiac K_ATP and L-type Ca²⁺ currents, cytosolic Mg²⁺, adenine nucleotide phosphates, and EC coupling
The whole-cell model was able to simulate quantitatively or qualitatively various experimental measurements in normal and pathological conditions and to make predictions that are possible to test experimentally. The most important theoretical result was our observation that the changes in normal diastolic free ATP, MgADP, and MgATP concentrations, as a consequence of the changes in absolute cytosolic Mg²⁺, ATP, and ADP levels regulate the K_ATP and L-type Ca²⁺ channel activity and the integrated process of excitation-contraction coupling in ventricular myocytes. In addition, model studies demonstrated (see also Michailova and McCulloch, 2001) that the changes in intracellular Ca²⁺ concentrations ( and) during cell excitation were not able to cause sensitive alterations in the diastolic free ATP, MgADP, and MgATP normal or pathological level.

Calculations showed that the current model quantitatively reproduces Nichols et al. (1991a) metabolic experiments on dependence of K_ATP channel activity in the presence of normal and However, the model was able only qualitatively to simulate the effects of metabolic blockade on and AP time courses during a single beat. The analysis suggests that the enhanced K_ATP current and markedly shortened current and AP-potential durations, were due to decreased free diastolic ATP and increased diastolic MgADP level. In addition, results revealed that the drop in total ATP inhibited Ca²⁺ signal and consequently contractile activity (in this article we assume contractile force f ). The analysis suggests that the most important reason for this was the downregulation of SERCA2a pump by the reduced diastolic MgATP levels. Here, we hypothesize that the experimentally recorded twitches (Nichols et al., 1991a) reach maximum amplitude slower than calculated because actin and myosin interactions as well MgATP regulation of myosin ATP-ase are not included in the model, yet. Finally, this model was able to simulate qualitatively the effects of rapid increase in intracellular diastolic MgATP on Ca²⁺ channel activity (O'Rourke et al., 1992; Yazawa et al., 1997).

An interesting feature of this model was its ability to simulate the modulation of ATP sensitivity of K_ATP channel by ADP in the presence of Mg²⁺. Our studies demonstrated that at all concentrations below 500 µM (with total Mg²⁺ normal), the increase in ADP stimulated channel activity. However, at all concentrations >500 µM, the increase in ADP caused channel inhibition. How could we explain these "paradoxical ADP effects" on the channel activity? The simulations here revealed that for < 500 µM the ratio was greater than unity and In contrast, for all > 500 µM the ratio was below unity and In addition, calculations showed that in both cases the diastolic MgADP increased and > 1. Therefore, we concluded that in cardiac myocytes, a high level of free ADP would be expected to cause channel inhibition when < 1 (or < ). Our predictions for the effects of ADP in modulating ATP sensitivity of K_ATP channel are in agreement with reported experimental data in rat and guinea pig myocytes (Lederer and Nichols, 1989; Weiss et al., 1992). To explain the observed "paradoxical ADP effect" Lederer and Nichols (1989) hypothesize that total ADP becomes greater than total Mg²⁺ with > 500 µM. Our studies showed that here <

This model is also able to predict how the changes in cytosolic ATP in the absence of SUR2A activation (ADP or Mg²⁺ 0 mM) or the changes in cytosolic ADP or Mg²⁺ in the absence of Kir6.2 inhibition (ATP 0 mM) regulate current. Computations showed that, in the absence of ADP and presence of Mg²⁺, the increase of total ATP (or ) significantly inhibited current. In contrast, in the absence of ATP and presence of Mg²⁺, the increase of total ADP (or ) sensitively activated current through the channel. These model predictions are in qualitative agreement with experimental data by Lorenz et al. (1998) in pancreatic K_ATP channels. However, we could not find experimental measurements in the literature for or regulation alone of cardiac current.

It is commonly expressed that excitability and contractility are determined by ratio. By testing this, we demonstrated that the absolute levels of adenine nucleotides regulate K_ATP and L-type Ca²⁺ channel activity and cardiac EC coupling independently of the ratio. Our model predicts that simultaneous decrease in and (with total Mg²⁺ unchanged) activate K_ATP current, do not affect peak, inhibit signal, and markedly shorten action potential, and current durations. The analysis suggests that observed changes in myoplasmic Ca²⁺ signal and were due to the changes in diastolic and whereas those in current due to the alterations in diastolic and Therefore, our data and those of others (Dunne et al., 1988; Albitz et al., 1990) reveal that understanding of the control of K_ATP and L-type Ca²⁺ channel activities and of the integrated EC coupling process requires knowing more metabolic variables than the ratio. New electrophysiological measurements in cardiac myocytes need to be done to test further the ratio hypothesis and the correctness of our model predictions.

Despite of the fact that Mg²⁺ is the most abundant divalent cation in the cell, little is known about intracellular Mg²⁺ homeostasis and mechanisms controlling [Mg²⁺]_i (Murphy, 2000). The absence of detectable major changes in [Mg²⁺]_i and the slower turnover of the cation across the cell membrane in normal conditions have supported for more than three decades the assumption that total Mg²⁺ content is kept constant at the level necessary for enzyme and channel function, and that its concentration need not change rapidly to form complexes with ATP and other phosphonucleotides. However, a body of new experimental results now suggests that large fluxes of Mg²⁺ can cross the cell membrane in either direction (via electroneutral Na⁺/Mg²⁺ exchanger or selective Mg²⁺ channels) following a variety of hormonal and nonhormonal stimuli and inducing sensitive changes in total Mg²⁺ and little or no change in free cytosolic Mg²⁺ (Romani and Scarpa, 1990, 2002; Fatholahi et al., 2000). Here our new model provided a unique opportunity for the first time to investigate theoretically how simultaneous variations in intracellular Mg²⁺ ( or ) may affect Ca²⁺ transients, sarcolemmal K_ATP current, and other ionic currents involved in action potential genesis. Computations showed that the changes in (respectively in total Mg²⁺), even around the reported range of physiological concentrations (0.2–1.8 mM) with total ATP and ADP normal, may have pronounced effect on on Ca²⁺ signals and the time course of the action potential. The analysis suggests that the main reason for the observed effects were the alterations in normal free ATP, MgADP, and MgATP levels. Our prediction that the increase in leads to shortening of action potential duration is in a qualitative agreement with experimental data by Agus et al. (1989). In addition, experimental data suggest that free Mg²⁺ and Mg²⁺-nucleotide complexes may exert opposite effects on L-type Ca²⁺ current, i.e., increases in MgATP activate whereas increases in inhibit the current (O'Rourke et al., 1992). Our calculations showed that the increase in increased peak. However, the model failed to predict that an increase in dramatically suppresses the L-type Ca²⁺ current (Agus et al., 1989; Pelzer et al., 2001; Yamaoka et al., 2002; Wang et al., 2003). Recent experimental studies by Wang et al. (2003) in rat cardiac myocytes suggest that the interaction between and to modulate is not significantly affected by ryanodine, fast Ca²⁺ buffers, or inhibitors of calmodulin, calmodulin-dependent kinase, and calcineurin. The authors concluded that physiologically relevant modulates by counteracting the effects of Ca²⁺ channel phosphorylation or by an unknown -dependent mechanism.

Block of oxidative metabolism and a fall in cause significant changes in ion concentrations () and have important effects on ion channels and carriers (Lederer and Nichols, 1989; Isenberg et al., 1993; Ch'en et al., 1998; Carmeliet, 1999; Bers, 2001). Weiss et al. (1992) report that, while creatine phosphate is present intracellular ATP and ADP remain unchanged but once ATP is depleted a large positive increase in free myoplasmic ADP is observed because the later arises from the hydrolysis of ATP. In this article we examined how the changes in total ATP and free ADP reported by Weiss et al. after 40s of 10 min ischemia might affect the normal cell excitability and contractility. Simulations with the model demonstrated that a fall in (with total Mg²⁺ normal) significantly reduces sarcoplasmic Ca²⁺ content, increases diastolic Ca²⁺, lowers systolic Ca²⁺, increases Ca²⁺ influx through L-type channels, and decreases the efficiency of the Na⁺/Ca²⁺ exchanger in extruding Ca²⁺ (Isenberg et al., 1993; Carmeliet, 1999). These simulations also resulted in a sensitive decrease of action potential duration, significant activation of ATP-sensitive K⁺ current, and an increase of free Mg²⁺ that has been experimentally observed during ischemia (Carmeliet, 1999). Our analysis suggests that the most important reasons for the observed changes during metabolic inhibition were: 1), the downregulation of SERCA2a pump activity by the reduced diastolic MgATP, and 2), the activation of current due to decreased free ATP and increased MgADP levels.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS DISCUSSION CONCLUSIONS APPENDIX I: MICHAILOVA-McCULLOCH... APPENDIX II: GLOSSARY ACKNOWLEDGEMENTS REFERENCES

 
Cytosolic free Mg²⁺ and cytosolic ATP, MgATP, ADP, and MgADP regulate a large number of cellular processes. The improved ionic-metabolic model allowed us to investigate how the changes in free and bound Mg²⁺, ATP, and ADP during cell excitation regulate the availability of ATP-sensitive K⁺ and L-type Ca²⁺ channels and the integrated process of excitation-contractio