GLOSSARY

TOP ABSTRACT GLOSSARY INTRODUCTION MATHEMATICAL MODEL METHODS AND MODEL PARAMETERS RESULTS DISCUSSION REFERENCES

Abbreviations

Volumes, areas, and capacity

Membrane currents

Concentrations

Fluxes

Jxfer	=	Ca2+ flux from subspace to myoplasm
Jrel	=	RyR channel Ca2+ flux
J*up	=	modified Ca2+ uptake into NSR by SR Ca2+-ATPase pump
Jtrpn	=	buffering of Ca2+ by troponin C
J	=	Mg2+ flux from subspace to myoplasm
J	=	CaATP flux from subspace to myoplasm
J	=	MgATP flux from subspace to myoplasm
J	=	CaADP flux from subspace to myoplasm
J	=	MgADP flux from subspace to myoplasm

Time constants

xfer	=	time constant for transfer of Ca2+ from subspace to myoplasm
	=	time constant for transfer of Mg2+ from subspace to myoplasm
	=	time constant for transfer of CaATP from subspace to myoplasm
	=	time constant for transfer of MgATP from subspace to myoplasm
	=	time constant for transfer of CaADP from subspace to myoplasm
	=	time constant for transfer of MgADP from subspace to myoplasm

Dissociation and rate constants

K	=	Ca2+-ATP dissociation constant
k	=	Ca2+ on-rate constant for ATP
k	=	Ca2+ off-rate constant for ATP
K	=	Mg2+-ATP dissociation constant
k	=	Mg2+ on-rate constant for ATP
k	=	Mg2+ off-rate constant for ATP
K	=	Ca2+-ADP dissociation constant
k	=	Ca2+ on-rate constant for ADP
k	=	Ca2+ off-rate constant for ADP
K	=	Mg2+-ADP dissociation constant
k	=	Mg2+ on-rate constant for ADP
k	=	Mg2+ off-rate constant for ADP

	INTRODUCTION

TOP ABSTRACT GLOSSARY INTRODUCTION MATHEMATICAL MODEL METHODS AND MODEL PARAMETERS RESULTS DISCUSSION REFERENCES

A number of mathematical models have been developed to investigate Ca²⁺ signaling in cardiac muscle cells (Robertson et al., 1981; Michailova and Spassov, 1992; Stern, 1992; Amstudz et al., 1996; Keizer and Levine, 1996; Langer and Peskoff, 1996; Negroni and Lascano, 1996; Soeller and Cannell, 1997; Jafri et al., 1998; Hunter et al., 1998; Nygren et al., 1998; Peskoff and Langer, 1998; Winslow et al., 1998, 1999; Dawson et al., 1999; Michailova et al., 1999; Rice et al., 1999; Zoghbi et al., 2000). However, these models cannot be used to predict how the binding and transport of Ca²⁺ and Mg²⁺ by the mobile buffers, ATP and ADP, or a fall in [ATP]_tot/[ADP]_tot ratio might modulate intracellular Ca²⁺, Mg²⁺, Na⁺, and K⁺ concentrations, ion pumps and currents, or, conversely, how changes in free Ca²⁺ concentrations during excitation could alter free and bound concentrations of ATP and ADP. Recently, ATP diffusion and Ca²⁺ and Mg²⁺ exchange with ATP have been introduced in models of smooth muscle cells (Kargacin and Kargacin, 1997) and skeletal muscle cells (Baylor and Hollingworth, 1998). Cardiac energetics (metabolism of high-energy phosphates, glycogen metabolism, and lactate transport) and pH regulation have also been integrated with electrophysiological models (Ch'en et al., 1997, 1998; Shaw and Rudy, 1997).

Winslow et al. (1999) modified the model of Jafri et al. (1998) for guinea pig ventricular cells to study the mechanisms of Ca²⁺ handling in the canine midmyocardial ventricular myocytes. This integrative model incorporated: 1) membrane ion currents from the Luo-Rudy phase II ventricular cell model (Luo and Rudy, 1994); 2) the formulation of Jafri et al. (1998) for the L-type Ca²⁺ current that exhibits the mode-switching behavior observed by Imredy and Yue (1994); 3) SR Ca²⁺ release from RyR channels described in the Keizer and Levine (1996) model with receptor adaptation; 4) a subsarcolemmal space; 5) Ca²⁺ buffering by low- and high-affinity Ca²⁺ binding sites on troponin, and Ca²⁺ buffering by calmodulin and calsequestrin. In the model of Winslow et al. (1999), calculated subsarcolemmal Ca²⁺ could reach high levels (~30 µM) during excitation and rose more rapidly than myoplasmic Ca²⁺ (~0.5-0.6 µM). It faithfully reproduced measured Ca²⁺ transients in normal and failing canine ventricular myocytes (O'Rourke et al., 1999).

In this study, we extend the model of Winslow et al. (1999) by incorporating equations for Ca²⁺ and Mg²⁺ buffering and transport by ATP and ADP and equations describing ATP (or MgATP) regulation of ion transporters (Na⁺-K⁺ ATPase pump, sarcolemmal Ca²⁺-ATPase pump, SR Ca²⁺-ATPase pump). Our results support the hypothesis that, under normal conditions, Ca²⁺ binding by low-affinity mobile buffer ATP and diffusion of Ca²⁺-bound ATP (CaATP) may contribute to the amplitude and time course of intracellular Ca²⁺ signals. The inclusion of ATP and ADP buffering slightly decreased the peak of the myoplasmic Ca²⁺ transient computed in response to periodic voltage-clamp stimuli. The addition of CaATP diffusion slightly decreased the peak and accelerated the relaxation of the Ca²⁺ transient in the subspace. Metabolic changes (a fall in [ATP]_tot/[ADP]_tot ratio) significantly reduced SR Ca²⁺ content, increased diastolic Ca²⁺ levels, and decreased systolic Ca²⁺ levels computed in response to periodic voltage-clamp stimuli. As a result, Ca²⁺ influx through L-type Ca²⁺ channels increased while the efficiency of Ca²⁺ extrusion by the Na⁺/Ca²⁺ exchanger decreased. The main reason for these changes was reduced Ca²⁺ uptake by the SR Ca²⁺-ATPase (SERCA2a pump) due to decreased diastolic MgATP levels during metabolic inhibition. In ventricular myocytes, high Ca²⁺ concentrations developed near the membrane might influence Mg²⁺, ATP, and ADP concentrations more significantly than the negligible alterations in these concentrations stimulated by low myoplasmic Ca²⁺.

	MATHEMATICAL MODEL

TOP ABSTRACT GLOSSARY INTRODUCTION MATHEMATICAL MODEL METHODS AND MODEL PARAMETERS RESULTS DISCUSSION REFERENCES

The overall scheme of the model is shown in Fig. 1. (See Glossary for the notations of the parameters used throughout the study.) We provide only the additional or modified equations necessary to include ATP and ADP as Ca²⁺ and Mg²⁺ buffers and transporters, and ATP as Na⁺-K⁺, sarcolemmal, and SR Ca²⁺ pumps regulator. The remaining equations were the same as those in the original paper of Winslow et al. (1999) with the corrections as given on the author's web site.

View larger version (48K): [in this window] [in a new window]   FIGURE 1   Schematic diagram of the mechanisms involved in the model.

Experimental data suggest that the SR is not accessible to the mobile buffers, ATP and ADP (Bers, 1991; Carmeliet, 1999). Therefore, in the model, ATP and ADP are free to react and diffuse within the subspace and bulk myoplasm but not in the SR (Fig. 1). It was also assumed that the total ATP and ADP concentrations in the subspace ([ATP]_totss, [ADP]_totss) and bulk myoplasm ([ATP]_toti, [ADP]_toti) are equal, spatially uniform, and remain constant during excitation, i.e.,

(1)

(2)

The buffering of Ca²⁺ and Mg²⁺ by ATP in the subspace is given by the following equations:

(3)

(4)

(5)

where k₊ and k are the corresponding on-rate and off-rate binding constants. The equations for the bulk myoplasm are similar:

(6)

(7)

(8)

CaATP and MgATP fluxes (Fig. 1) are given by

(9)

(10)

For the subspace, adjustment of ATP fluxes by a factor of (V_myo/V_ss) is necessary to account for the different volumes of myoplasm (V_myo) and subspace (V_ss).

Similar equations for the Ca²⁺ and Mg²⁺ exchange with ADP in the subspace and bulk myoplasm can be written as

(11)

(12)

(13)

(14)

(15)

(16)

CaADP and MgADP fluxes (Fig. 1) are given by

(17)

(18)

In the model, we assume that the transfer of free and bound ATP and ADP from the subspace to the myoplasm occurs at the same rate. The changes in free Mg²⁺ concentration during excitation in the subspace are described by

(19)

and in the bulk myoplasm by

(20)

The transfer flux of Mg²⁺ from subspace to myoplasm (Fig. 1) is given by

(21)

ATP and ADP not only buffer and transport Ca²⁺ and Mg²⁺ ions but also have well-known regulatory functions in the cell (Bers, 1991; Leyssens et al., 1996; Carmeliet, 1999). In cardiac myocytes, ATP (as MgATP) drives a number of enzymes, channels (ATP-sensitive K⁺ channels), and transporters (Na⁺-K⁺ ATPase pump, sarcolemmal Ca²⁺- ATPase pump, SR Ca²⁺-ATPase pump) (Noma, 1983; Fozzard and Lipkind, 1995; Shaw and Rudy, 1997; Yokoshiki et al., 1998; Carmeliet, 1999). To simulate transporter ATP regulation, we modified the equations of Winslow et al. (1999) for Na⁺-K⁺ pump current (I_NaK), sarcolemmal Ca²⁺ pump current (I_p(Ca)), and SR Ca²⁺ ATPase pump (J_up):

(22)

(23)

(24)

where

(25)

and [MgATP]_i0 is the resting myoplasmic MgATP concentration in normal conditions ([ATP]_tot = 7 mM, [ADP]_tot = 5 µM, free Mg²⁺ = 1 mM).

The equations for the [Ca]_ss and [Ca]_i in the model of Winslow et al. (1999) were modified, taking into account that now ATP and ADP buffer Ca²⁺ and regulate ion pumps:

(26)

(27)

where _ss is a rapid buffering approximation factor for calmodulin in the subspace, and _i is a rapid buffering approximation factor for calmodulin in the myoplasm.

	METHODS AND MODEL PARAMETERS

TOP ABSTRACT GLOSSARY INTRODUCTION MATHEMATICAL MODEL METHODS AND MODEL PARAMETERS RESULTS DISCUSSION REFERENCES

The system of first-order nonlinear differential equations at given initial conditions was solved using Gill's modification of the Runge-Kutta fourth-order algorithm (Ralston and Wilf, 1960). The maximum step size for time integration was 0.1 ms and the maximum error tolerance was 10⁶.

Total ATP and ADP concentrations used in the model (see Table 1) are average values measured in different cardiac tissues and species (Bers, 1991; Leyssens et al., 1996; Baylor and Hollingworth, 1998; Ch'en et al., 1998). ATP and ADP dissociation constants (see Table 2) were taken from Martell and Smith (1982), and Kargacin and Kargacin (1997). The on- and off-rate constants, k and k, were obtained from Baylor and Hollingworth (1998). We could not find published data for the values of CaADP, MgATP, and MgADP on- and off-rate binding parameters. Therefore, the typical near-diffusion-limited on-rate value of 125 µM¹ s¹ has been assumed (see Table 2) (Soeller and Cannell, 1997; Ch'en et al., 1998). The corresponding off-rate constants were obtained from the known values of equilibrium dissociation constants (K, K, K).

It is known that free Mg²⁺ in cardiac cells is between 0.5 and 2 mM (Bers, 1991; Leyssens et al., 1996; Carmeliet, 1999; Murphy et al., 1989). Here we assume that, at equilibrium, free Mg²⁺ concentrations in the subspace and myoplasm do not differ, a value of 1 mM was used for both (see Table 3). Total Mg²⁺ concentration (~7.44 mM) was estimated from the accepted free Mg²⁺ concentration of 1 mM at ~100 nM free Ca²⁺. We also assumed that total Mg²⁺ concentration in the cell remains constant, while free Mg²⁺ concentration varies with concentration of total ATP and ADP as Ca²⁺ and Mg²⁺ buffers. The Mg²⁺ transfer-time constant from the subspace to the myoplasm (see Table 4) was chosen to be equal to the Ca²⁺ transfer-time constant (_xfer) used by Winslow et al. (1999). The transfer of free and bound ATP and ADP was assumed to occur at half the rate of free Ca²⁺ and Mg²⁺ diffusion (see Table 4) (Baylor and Hollingworth, 1998). Unless specified otherwise in the figure legends or in the text, the standard set of parameters and initial conditions used in the calculations is listed in the Tables. All other initial conditions and values of the parameters that are not included in the Tables correspond to those listed in the Appendix of Winslow et al. (1999).

In this paper, the effects of ATP and ADP buffering, transport, and regulation in normal conditions and during metabolic inhibition were examined in response to periodic voltage-clamp stimuli (97 mV holding potential, 3 mV step potential, 200 ms duration) at a frequency of 1 Hz.

	RESULTS

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Ion and buffer concentrations and ion currents in normal conditions

Ca²⁺, Na⁺, K⁺ concentrations and ion currents with inclusion of Mg²⁺, ATP, and ADP

View larger version (19K): [in this window] [in a new window]   FIGURE 2   Model outputs in response to 1-Hz periodic voltage-clamp pulse of 97 mV holding potential, 3 mV step potential, and 200 ms duration. Only responses to the 10th stimulus (9-10 s) are shown. (A) Subspace Ca2+ transient. (B) Myoplasmic Ca2+ transient. (C) JSR Ca2+ transient. (D) L-type Ca2+ current. Mg2+, ATP, and ADP not included (solid line). ATP and ADP treated as stationary buffers (dashed line). CaATP flux or all fluxes included (dotted line). For simulations (dashed and dotted lines) [ATP]tot = 7 mM, [ADP]tot = 5 µM, free Mg2+ = 1 mM. In all simulations shown, [Mg]tot = 7.44 mM, [K]0 = 4 mM, [Na]0 = 138 mM, [Ca]0 = 2 mM.

Free and bound Mg²⁺, ATP, and ADP concentrations in the presence and absence of fluxes

View larger version (10K): [in this window] [in a new window]   FIGURE 3   (A) Effect of changes in subspace Ca2+ concentration on free subspace ATP concentration and (B) of changes in myoplasmic Ca2+ concentration on free myoplasmic ATP concentration. ATP and ADP treated as stationary buffers (dashed line). All fluxes included (dotted line). Only responses to the 10th voltage-clamp stimulus (97 mV holding potential, 3 mV step potential, 200 ms duration, frequency 1 Hz) are shown. [ATP]tot = 7 mM, [ADP]tot = 5 µM, free Mg2+ = 1 mM.

View larger version (10K): [in this window] [in a new window]   FIGURE 4   Time courses of MgADP (9-10 s) in (A) subspace and (B) myoplasm. ATP and ADP treated as stationary buffers (dashed line). All fluxes included (dotted line). Simulations are generated in response to 1-Hz voltage-clamp pulse (97 mV holding potential, 3 mV step potential, 200 ms duration). [ATP]tot = 7 mM, [ADP]tot = 5 µM, free Mg2+ = 1 mM.

Ion and buffer concentrations, and ion currents during metabolic inhibition

Experimental studies have demonstrated that block of oxidative metabolism and a fall in [ATP]_tot/[ADP]_tot ratio cause important changes in ion concentrations ([K⁺]₀, [H⁺]_i, [Na⁺]_i, [Ca²⁺]_i, [Mg²⁺]_i) and have important effects on channels and carriers (Murphy et al., 1989; Marban et al., 1990; Wagner et al., 1990; Isenberg et al., 1993; Wilde and Aksnes, 1995; Carmeliet, 1999; Huser et al., 2000). Ch'en et al. (1998) reported that, while creatine phosphate is present, total ATP and ADP concentrations remain constant, at ~7 and ~0 mM, respectively. Once ATP is depleted, ADP starts to increase, and, after ~460 s, total ATP and ADP levels become approximately equal (~3 mM) during ischemia. Taking these results into consideration, we calculated ion and buffer concentrations and simulated ion currents when [ATP]_tot and [ADP]_tot were 3 mM and mobile. Initial free Mg²⁺ concentrations (t = 0 s) in the subspace and myoplasm ([Mg]_ss, [Mg]_i) were estimated to be ~2.248 mM, assuming that total intracellular Mg²⁺ concentration remains constant (7.44 mM) after the fall in [ATP]_tot/ [ADP]_tot ratio from 1400 (normal conditions) to unity. During these simulations, the initial free and bound metabolic ATP and ADP concentrations were [ATP]_ss = 112 µM, [CaATP]_ss = 0.05 µM, [MgATP]_ss = 2.888 mM, [ATP]_i = 112 µM, [CaATP]_i = 0.047 µM, [MgATP]_i = 2.888 mM, [ADP]_ss = 694 µM, [CaADP]_ss = 0.045 µM, [MgADP]_ss = 2.3 mM, [ADP]_i = 694 µM, [CaADP]_i = 0.038 µM, and [MgADP]_i = 2.3 mM. Under metabolic conditions for the period of 10 s stimulation, the extracellular concentrations of K⁺, Na⁺, and Ca²⁺ were kept constant at normal values ([K]₀ = 4 mM, [Na]₀ = 138 mM, and [Ca]₀ = 2 mM). All other initial conditions and values of the parameters used were those listed in our Tables and in the Appendix of Winslow et al. (1999).

Ca²⁺, Na⁺, K⁺ concentrations and ion currents

View larger version (13K): [in this window] [in a new window]   FIGURE 5   Model outputs in response to 1-Hz periodic voltage-clamp pulse of 97 mV holding potential, 3 mV step potential, and 200 ms duration. (A) Subspace Ca2+ transient (9-10 s). (B) Myoplasmic Ca2+ transient (9-10 s). (C) JSR Ca2+ transient (9-10 s). ATP and ADP are treated as mobile Ca2+ and Mg2+ buffers. [ATP]tot = 7 mM, [ADP]tot = 5 µM, free Mg2+ = 1 mM (dotted line). [ATP]tot = 3 mM, [ADP]tot = 3 mM, free Mg2+ = 2.248 mM (solid line). [ATP]tot = 0 mM, [ADP]tot = 0 mM, free Mg2+ = 7.44 mM (dash-dot line). Top traces in (A), (B), and (C) show [Ca]ss, [Ca]i, and [Ca]JSR transients during metabolic inhibition ([ATP]tot = [ADP]tot = 3 mM, free Mg2+ = 2.248 mM) for the period 0-11 s.

View larger version (26K): [in this window] [in a new window]   FIGURE 6   Model simulations for a periodic voltage-clamp pulse of 97 mV holding potential, 3 mV step potential, 200 ms duration, frequency 1 Hz. (A) L-type Ca2+ current. (B) Na+-Ca2+ exchange current. (C) Na+-K+ pump current. (D) Sarcolemmal Ca2+ pump current. ATP and ADP are treated as mobile Ca2+ and Mg2+ buffers. [ATP]tot = 7 mM, [ADP]tot = 5 µM, free Mg2+ = 1 mM (dotted line). [ATP]tot = 3 mM, [ADP]tot = 3 mM, free Mg2+ = 2.248 mM (solid line). [ATP]tot = 0 mM, [ADP]tot = 0 mM, free Mg2+ = 7.44 mM (dash-dot line). Top traces in (A), (B), (C), and (D) show ICa, INaCa, I*NaK, and I*p(Ca) with metabolic inhibition ([ATP]tot = [ADP]tot = 3 mM, free Mg2+ = 2.248 mM) for the period 0-11 s.