J. theor. Biol. (1992) 156, 443-483

Sodium-Calcium Exchange: Derivation of a State Diagram and Rate Constants from Experimental Data EDWARD A. JoItNSONaf, D. RENALD LEMIEUXJ~ AND J. MAILEN KOOTSEY§

t Department of Cell Biology, Box 3709, Duke University Medical Center, Durham, NC 27710, U.S.A. ~ Institut de g~nie biomedical, Facult~ de M~dicine, Universit~ de Montrkal, C.P. 6128 Succ. A, Montreal, Quebec, Canada H 3 C 3 J 7 and § Universi O, College of Arts and Sciences, Andrews UniversiO,, Berrien Springs, MI 49104, U.S.A. (Received on 5 November 1990, A ccepted on 26 December 1991 ) A mechanism is developed for Na+-Ca 2÷ exchange using a new approach made possible by the availability of computer software that allows the systematic search of a large parameter space for optimum sets of parameters to fit multiple sets of experimental data. The approach was to make the experimental data dictate the form of the mechanism: the qualitative features of the data dictating the number and nature of the states of the exchanger and their interrelationship, and the quantitative aspects of the data dictating the values of the rate constants that govern the amount of each state relative to the total amount of exchanger. A single set of experimental data served this initial purpose, namely, observations of equilibrium Ca2÷-Ca 2+ exchange in cardiac sarcolemmal vesicles (Slaughter et al., 1983, J. biol. Chem. 258, 3183-3190). From this data a minimum mechanism was induced having 56 states (SYM56), which gave satisfactory quantitative fits to the experimental data. With this set of parameters additional experimental data were fitted, from the same preparation, the single cardiac cell and the squid giant axon, with some changes in parameters, but none dramatic. In spite of the symmetric nature of the mechanism, i.e. binding constants for Na ÷ and Ca 2+ do not depend on the orientation of the binding sites, the mechanism exhibits marked asymmetric behavior similar to that observed experimentally. Finally, in accounting for Ca2~--Ca2+ exchange in the absence of monovalent cations, Ca 2+ influx becomes dependent on intracellular Ca2+--an unexpected outcome--exactly in keeping with the "essential activator" role of intracellular Ca 2~ observed by DiPolo & Beaug6 ( 1987, J. gen. Physiol. 90, 505-525). Observations of Na+-Ca 2+ exchange in the retinal rod outer segment are well fitted with a simplified version of SYM56 comprising 25 states (namely, SYM25), supporting the notion that the exchanger in the retinal rod outer segment differs from that in cardiac sarcolemma and squid axon. Maximum turnover rate of 840 sec -~ for SYM56 and 20 sec -~ for SYM25 are comparable to those reported for the exchanger in cardiac muscle and retinal rod outer segment, respectively.

I. Introduction The crucial role o f N a + - C a 2+ exchange in ion homeostasis and the need for numerical models o f the exchanger as essential tools for exploring the complexities o f ion homeostasis has been eloquently expressed in a recent review article by Hilgemann (1988) o f numerical a p p r o x i m a t i o n s o f N a + - C a 2+ exchange. Part o f the b a c k g r o u n d 443

0022-5193/92/I 20443 + 41 $03.00/0

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to his review was a "two-step simultaneous" mechanism for Na+-Ca 2÷ exchange described by Johnson & Kootsey (1985), in which three Na ÷ were exchanged, simultaneously, for a single Ca 2+. The mechanism was asymmetric in that there were only four binding sites: three Na+-specific sites facing one side and one Ca2+-specific site facing the other side of the membrane. On the grounds of mechanistic simplicity, ion-binding to these sites was assumed to be random and, in accord with experimental finding (Philipson & Nishimoto, 1982; Philipson, 1985), the binding constants for Na + and for Ca 2+ were unaffected by the orientation of the sites in the membrane. The behavior o f this mechanism, named SIMVI I, was found to be consistent with a broad range of experimental observations of Na +-Ca 2+ exchange in cardiac muscle, the squid giant axon and isolated sarcolemmal vesicles (Johnson & Kootsey, 1985). The initial purposes of our present study were two-fold: to modify S I M V l l to account for monovalent cation-activated Ca2+-Ca 2+ exchange, a behavior of the exchanger that was not addressed in the original formulation of SYMVII and, secondly, to test the ability of SIMVI 1 to account for new experimental data (e.g. the voltage-dependency of Na+-Ca 2+ exchange). I n the course of this study, we found that the Ca2+-Ca 2+ exchange exhibited by SIMVI 1 was fundamentally different from that observed experimentally. Such discrepant behavior proved to be an inherent property of asymmetric mechanisms such as SIMVII and we were therefore forced to abandon it and search for an alternative mechanism. Our approach differed from the one that we used to develop the previous mechanism, SIMVI 1. As will be seen later, this new approach was made feasible by the availability of computer software that allowed the systematic search of a large parameter space for optimum sets of model parameters to fit multiple sets of experimental data. The approach was to make the experimental data dictate the form of the mechanism : the qualitative features of the data dictating the number and nature of the states of the exchanger and their interrelationships, and the quantitative aspects of the data dictating the values of the rate constants that govern the amount of each state relative to the total amount of exchanger. The result is two exchange mechanisms: one that accounts for both Ca2+-Ca 2+ exchange and Na+-Ca z+ exchange in cardiac muscle and the squid giant axon, and one of simpler form that accounts for Na+-Ca 2+ exchange in the retinal rod outer segment. An unplanned and totally unexpected property of the former mechanism is that Ca 2+ influx depends on intracellular Ca2+--exactly in keeping with the "essential activator" role o f intracellular Ca 2+ observed by DiPolo & Beaug6 (1987).

2. Preliminary Considerations 2.1. F A I L U R E

OF THE ASYMMETRIC

SIMULTANEOUS

MECHANISM

SlMVII

As we discovered, there are several sets of experimental observations that cannot be accounted for by the exchange mechanism S l M V l l , the state diagram of which is shown in Fig. l(a). We will describe only one such set since this set is all that is necessary to justify abandoning SIMVI I. Moreover, as will be seen, the complete

N a + - C a 2+ E X C H A N G E

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445

reaction diagram of the exchange mechanism can be induced from this one set of observations. The set of experimental observations is that of Slaughter et al. (1983) who studied the influence of monovalent cations on Ca2÷-Ca 2÷ exchange in cardiac sarcolemmal vesicles under equilibrium conditions. Such conditions ensured no net transport of either the monovalent cation or Ca 2+ and therefore stability in their respective concentrations. Nonetheless, the unidirectional, isotopic, fluxes of Ca z÷ associated with equilibrium Ca2÷-Ca 2÷ exchange could be measured at various concentrations of the monovalent cations. In one set of experiments, the initial rate of 45Ca2+uptake was measured in vesicles equilibrated with a monovalent cation-free (sucrose) solution containing 100 laM Ca 2÷ (for the measurement, the extravesicular Ca 2÷ was replaced with aSCa2+). When the sucrose was replaced (both intra- and extravesicularly) by the same concentration of a monovalent cation, such as K ÷, the exchange increased, the increase reaching a maximum plateau value at around 15 mM, about twice the value in sucrose alone. On the other hand, when Na ÷ was the replacement cation, the exchange increased to a maximum at around 5 mM, again about twice that in sucrose alone. Beyond 5 mM, however, the exchange steeply declined to negligible values at 100 mM and greater. This inhibitory effect on Ca2÷-Ca 2÷ exchange was peculiar to Na ÷ and was not exhibited by high concentrations of other monovalent cations, such as Li ÷ and Rb ÷. As the reaction scheme stands [see Fig. I(a)] S1MVll cannot exhibit Ca2÷-Ca 2÷ exchange in the absence of Na ÷ since Ca2+-Ca ~÷ exchange can only occur via the fully loaded states, Na3ECa and Na3E'Ca. Nevertheless, this behavior can be introduced into SIMV11 by adding additional states, similar to those in the new mechanism, to be described below. However, with these additions and under equilibrium conditions mimicking those of Slaughter et al. (1983), increasing concentrations of Na ÷ replacing sucrose, cause Ca2+-Ca 2÷ exchange to increase to a maximum plateau from which it fails to decline, a behavior characteristic of that for other monovalent cations, such as for K ÷, described above, not that for Na ÷. As will become clear later when the new mechanism is discussed, such aberrant behavior is inherent in asymmetric mechanisms such as the simultaneous one of SIMVI I.

2.2. SIMULTANEOUS VS. CONSECUTIVE MECHANISM

As pointed out previously (Johnson & Kootsey, 1985), a purely consecutive mechanism, as exemplified by that proposed by L/iuger (1987), is excluded simply on the basis of the experimental observation that Ca2+-Ca 2÷ exchange in the absence of any monovalent cation is quite small compared to that seen in the presence of Na+. For example, DiPolo & Beaug6 (1981) observed a vanadate-insensitive Ca 2÷ efflux in the absence of Na ÷ that was around 7% of that seen in its presence. It can be readily recognized from inspection of the reaction diagram for a consecutive mechanism [see Fig. l(b)] that the rate of Ca2÷-Ca 2+ exchange in the absence of Na ÷ would be greater than that in its presence since, in the absence of Na ÷, the amount of Ca 2+bound states of the exchanger must necessarily be larger when the Na+-bound states

[

Na~

E . N a . ~ . ~ ~

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Ca~÷-- ~ ~ C a+~ ~

E.Naz ~

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No3ENo Nap* Na.+No2 ENOa No,.+ No + NoENo3 NoaENOz

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FIG. 4. SYM56. Fiftymsix-state N a 4 - C a 2+ exchange d i a g r a m derived from experimental o b s e r v a t i o n s o f equilibrium C a 2 * - C a 2~ exchange in cardiac sarcolemmal vesicles [cf. Slaughter et al. (1983: fig. 2)]. The stoichiometry of the exchanger is 3 N a ' : 1 Ca 2+. The exchanger has seven ion-free states: E, e, & X, • + 21X, x and x. Stmultaneous Na -Ca exchange occurs through the translocating conformational change: CaENa3*--,Na3ECa. Consecutive N a ' -Ca 2÷ exchange occurs through the states X and x, .~ and 2, E and e, E and #, see Fig. 5. The voltage-dependent steps are the Ca z+ translocational steps: Na)ECa*-* CaENa3, CaE~-.-~Ca6, ECa*--,eCa, XCa~-,xCa, and Ca.~*-+ Ca.~. The equilibrium constant for the translocational exchange step of each of the five exchange cycles is equal to exp ( F V / R T ) (see text). The translocational steps are indicated with an asterisk. - -

-

and CaENa3. Furthermore, net consecutive N a + - C a z+ exchange can occur through Na ÷ and Ca 2+ forms of the states e, ~, X, x, .I~ and ~. There are four such consecutive pathways, two involving the M+-free and two the M+-bound states, as shown diagramatically in Fig. 5. 3.2. M E C H A N I S M

FOR THE RETINAL ROD OUTER SEGMENT

The available experimental data from the retinal rod outer segment concerns only net N a + - C a 2+ exchange, there being no evidence for Ca2+-Ca 2+ exchange in the absence of Na ÷, or the activation of such exchange by monovalent cations other than Na +. This being the case, our approach in accounting for this data was to simplify SYM56 by removing those states that were demanded by Ca2+-Ca 2+

452

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FroG. 5. C o n s e c u t i v e N a ' - C a 2~ exchange pathways of SYM56. The numbers I-4 identify the translocation steps for each cycle. The translocational steps are indicated with an asterisk.

C a '~

exchange, thereby reducing the number of states to 25, as shown in Fig. 6. The question now arose as to whether to go ahead and change the stoichiometry to 4 N a ÷ : l Ca2+:l K + in accord with the findings of Cervetto et al. (1989). In the absence of any evidence for order in the binding of the 4 Na, 1 Ca 2÷ and i K ÷, this would require an additional 40 states to SYM25 giving SYM65 (and a shift in the single translocational step from NasECa~--~ CaENa3 to Na4ECaK*--,KCaENa4). We decided otherwise, preferring first of all to explore whether SYM25 would fit the data of Lagnado et al. (1988). If SYM25 proved satisfactory, as turned out to be the case (see section 5), we reasoned that we could safely presume that SYM65 would also prove satisfactory. 3.3, V O L T A G E

DEPENDENCE

O F N a + C a 2+ E X C H A N G E

There is considerable experimental evidence from both cardiac muscle (e.g. Kimura et al., 1986) and the retinal rod outer segment (Hodgkin et al., 1987) showing clearly that N a ÷ - C a 2~ exchange is electrogenic. For example, Hodgkin et aL (1987) found that on average each Ca 2+ that was extruded from the retinal rod outer segment was accompanied by one single electron charge, or a little less. Given that the stoichiometry of the exchange is 3 Na ÷ for 1 Ca z÷ in cardiac muscle, as demonstrated by Reeves

N a + - C a 2+ E X C H A N G E

453

MECHANISM

CoECe

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coo,+ No.ECo /

co,,. No+ NozENOo+

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+ NaaENoz

Na3EN°z

N ° o + No, +

ENoz

[/Noo

~ CoENo~

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+ No + N°EN°3

NazENa3

No3ENa3 * FIG. 6, SYM25: retinal rod outer segment exchanger. Twenty-five-state, simultaneous N a ' - C a 2' exchange m e c h a n i s m for the retinal rod outer segment. The equilibrium constant for the one translocating step, Na3ECa~---~CaENa.~,was equal to exp (FV/RT) (see text). The binding constant for N a ' , namely K ~ . , is the same for all o f the 21 states of the exchanger to which it could bind. Similarly, the binding constant for Ca 2" is the same for all seven states of the exchanger to which it could bind, namely K~., T h e translocational step is indicated with an asterisk.

& Hale (1984), we have concluded that the charge transfer most likely involves the uncompensated exchange of 3 Na + for 1 Ca 2+, and we have also taken this to be the case for squid axon. In tile retinal rqd outer segment the stoichiometry appears to be 4 N a + : l Ca2+: l K + (Cervetto et al., 1989) however, as discussed above, for simplicity we employed the stoichiometry of 3 Na + : I Ca 2+. As a consequence, it is required thermodynamically that the overall equilibrium constant for each of the five exchange cycles in SYM 56 and the single cycle ill SYM25 (and in SYM65) be a function of the transmembrane potential. How this requirement was satisfied is discussed in the following section 3.5.

3.4. B I N D I N G C O N S T A N T S F O R Na + A N D Ca 2+

Philipson (1985) found that the Ko.s for activation of Na+-Ca 2+ exchange in isolated sarcolemmal vesicles by Na + was the same for an inside-out population of

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vesicles as it was for a mixed population of inside-out and right-side-out vesicles. The same was true for the activation of N a + - C a 2+ exchange by Ca 2+. As mentioned above, in the absence of experimental evidence to the contrary, for both SYM56 and SYM25 (SYM65) we have assumed random independent binding o f N a + and Ca 2÷ to the normal, ion-free state of the exchanger, E. T h a t is to say, the binding constant for Na + is the same for all of the 21 E states of the exchanger to which it can bind (cf. legends to Figs 4 and 6), namely, KN,,. e Similarly, the binding constant for Ca 2+ was taken to be the same for all of the nine E states to which it can bind (see legends to Figs 4 and 6), namely, K~a. In this way, binding of N a ÷ or Ca -,+, to one side of the exchanger has no effect on their binding to the other side. In keeping with the simplifying assumption of the independence of ion binding, for SYM56 we assumed that the binding constants for N a + to X and to )( were the same and were identical to that for Na ÷ to E, namely, Kr~a. Similarly, the binding constants of Ca 2+ to X and )( were assumed to be the same and identical to that of Ca 2+ to E, namely, KcF:~. On the other hand, there is no reason to assume that Na + binds with equal affinity to e, ~, x, and £c as it does to E, X and ,~. The same applies to the binding of Ca 2+. These states are derived from translocational, conformational changes in their parents, E, X and )(, so that the Ca 2÷ and Na + binding affinities could change in the process. It is reasonable to assume, however, that the binding affinities to e and to ~ be the same, as must be the case for the states x and 2, since the conformational change that gave rise to each must have been the same, the only difference being the side to which tile Ca 2+ or Na + bound. As a consequence, we have four additional binding constants: two more for Ca- , one for e and & Kc~, and one for x and .-% K~,,, and two more for Na +, one for e and (2, K~a and one for x and S:, K ~ , . This gives a total of six binding constants for Ca 2+ and N a +, as listed in Table I. Finally, we assumed a single binding constant for the monovalent cation, M +, namely K,~, identical for all states of tile exchanger to which it can bind. There was no justification for thinking that the binding affinity for M + to the ion-free state E should be any different from that for states in which the opposite side has bound a Ca 2+, one or more Na*, or for that matter, a M +. Hence, as listed in Table I, KM applies to the eight states to which it can bind: CaE, ECa, NaE, NazE, Na3E, ENa, ENa2 and ENa3. 3.5. E Q U I L I B R I U M C O N S T A N T S F O R T H E T R A N S L O C A T I O N A L

STEPS

Thermodynamics requires that, in the absence of a transmembrane potential, the overall equilibrium constant for each of the exchange cycles resulting in net N a ÷Ca 2. exchange must be unity. The symmetry in the binding constants for Na ÷ and Ca 2' to the E state of the exchanger causes the equilibrium constant for all ions binding to this state to be unity. For SYM25 and for the simultaneous reaction cycle of SYM56, the above thermodynamic requirement is ensured by assigning a value of unity to the equilibrium constant of the translocational step: Na3ECa~--~CaENa3. In SYM56 there are, in addition, four consecutive net exchange cycles, two involving E, e and E, ~, and two involving X, x and )(, :L Each cycle has two translocational

N a + - C a 2+ E X C H A N G E

MECHANISM

455

TABLE ! Symbol ( I) (2) (3) (4) (5) (6) (7)

K~, K~ K~:~ K~, K~, K~° K~

Units I molI mol-' I tool-' I tool-' I mol-' I tool -I I tool-'

Parameter description

(8) R,~

see-'

(9) RT~c' ( I 0) R~'c' (11) /~t~'

see -I sec- i sec-'

Binding constant of Ca 2~ to E, X and X Binding constant of Ca 2÷ to e and Binding constant o f Ca 2+ to x and _~ Binding constant o f N a ÷ to E, X and .,Y Binding constant of Na ~ to e and Binding constant o f Na + to x and .~ Binding constant of M 4 to E, ECa and CaE Rate constant for translocational steps Na3ECa,,--, CaENa3 Rate constant for translocational steps eCa*--, ECa and Ca~*--, CaE Rate constant for translocational steps xCa ~ X Ca and Ca.~,--, CaX Rate constant for translocational steps eNa3 ,---,ENa~ and

(12) R~I,~'

see -I

Rate constant for translocational steps xNa3,--.XNa3 and

Na3~ 4--,Na~E Na35 ~ NasX

(13) K ~ '

Equilibrium constant for translocational steps eCa~--bECa and

(14) K~'

Equilibrium constant for translocational steps xCa*-*XCa and

(15) K~,~'

Equilibrium constant for translocational steps eNa~ ~ ENa3 and

(16) K~"

Equilibrium constant for translocational steps xNa3*-+XNa~ and

Ca~ ~--~Ca E CaSe *-* CaX Na3~*-* Na3E Na3.~ ~ Na3X

(17) R~.~."

(I 8) E,o,.)

I mol- ' sec -t

Association rate for ion binding Total amount of the exchanger

steps, one for Na + and one for Ca 2+, giving a total of eight translocational steps. F o r simplicity, we have assumed that the equilibrium constants for the N a + translocational steps, XNa3*--,xNa3 and Na3.~Na3.~, were identical, namely, Keq '~" , a s w e r e those o f steps ENa3+-*eNa3 and Na3E~--~Na3~, namely K~". A similar set of assumptions was made for Ca 2+, i.e. the equilibrium constants for the translocational steps, XCa,--~xCa and CaR4--~Ca~,were identical, namely, K~q, .,-c, as were those of steps ECa~--~eCaand CaE*-+Ca~,namely, KcqeC". In each o f these exchange cycles, there are two binding constants for Na + and two for Ca 2+, as described above. Therefore, to ensure an overall equilibrium constant of unity for each o f the four consecutive cycles it was required that: K ~ . (K~q~)3 (K~,)3

(1)

K~. (K~) 3 -~N,_ .,c~. Keq -Kcq K~a. ( K ~ ) 3'

(2)

Ke'~'=K~'K*c~ and

3.6. A S S I G N M E N T OF RATE CONSTANTS

The dissociation rate constants for ion binding were calculated from the binding constants and an association rate constant that was assumed to be the same for all

456

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sites with an upper limit of 5 x 108 M-~ sec -I. A lower limit for the binding constant of 0-05 M-I was therefore imposed so that the dissociation rate could not exceed the diffusion rate limitation of 10 ~° sec -~. Table I lists all the parameters that define the rate constants of the reaction steps of SYM56 and SYM25, together with density of the exchanger (e.g. mol cm -2 of membrane or mol mg-~ of vesicular protein). Values for them were determined from best-fits to experimental data obtained by systematic searches of parameter space using software described below in section 4.

4. Methods

4.1. USE OF CONFORMATIONAL STATE THEORY It is appropriate to respond here to anticipated questions about our use of conformational state theory rather than other approaches that have been used to describe the movement of ions across membranes, notably, correlation analysis (Liebovitch & Fischberg, 1986), fractals (Liebovitch et al., 1987), and continuum theory (Levitt, 1990). These methods have been used successfully to describe the kinetics of single channel currents from patch-clamp experiments in terms of molecular events. Continuum theory, as defined by Levitt (1990) and the use of fractals are quite appropriate for describing ion channel kinetics because ions cross the membrane down their electrochemical gradient and ion-ion and ion-solution interactions, which these theories effectively describe, are key events controlling the motion of ions in the channel. In the processes of membrane transport and of exchange, however, the key events are quite different. At no time during the process of transport or exchange is there a direct connection between the two sides of the membrane, a key feature in the behavior of channels. In active ion transport there is a tight coupling between the energy released in the hydrolysis of ATP and the transmembrane electrochemical potential of the transported ions. In the case of exchange, there is a tight coupling between the transmembrane electrochemical potentials of the exchanging chemical species. That is to say, except in the process of substrate binding, no process analogous to diffusion occurs during ion transport and exchange. Contrary to ionic channels, where the transmembrane electrochemical potential energy of the "transported" ions is dissipated in the process of channel conductance, the electrochemical energy is conserved during its transfer within the transporter or exchanger protein. In the case of the ion transport ATPases, e.g. the various states of the enzyme protein associated with such energy transfer have been identified and the rate constants governing their interchange have been measured. Conformational state diagrams are therefore quite appropriate for describing the transport/exchange cycle, especially when the experimental observations are confined entirely to the macroscopic world, as they are in this paper. On the other hand, the experimental observations of ion channels are of a single channel-protein molecule modulating the flux of ions, a situation that demands an alternative to the macroscopic theory of conformational states that we employ here.

N a + - C a 2+ E X C H A N G E

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4.2. S T E A D Y - S T A T E E Q U A T I O N S D E S C R I B I N G T H E E X C H A N G E R E A C T I O N D I A G R A M

The two sets of linear algebraic equations relating the steady-state amount of each of the states comprising SYM56 and SYM25 (cf. Figs 4 and 6) relative to the other states, in terms of rate constants of the individual steps and the internal and external concentrations of Na +, C a 2+ (and M + in SYM56), were solved numerically. Solutions were obtained by Gaussian elimination employing the simulation software SCoP (Simulation Resources, 300 S. Bluff, Berrien Springs, MI 49103), running on a Sun-4/280 Workstation (Sun Microsystems, Inc., Mountain View, CA, U.S.A.). 4.3. C A L C U L A T I O N O F U N I D I R E C T I O N A L ION F L U X E S

The calculated unidirectional ion fluxes correspond to experimental measurements of isotopic ion flux, e.g. the efflux of 4SCa2+ from intracellular to extracellular space, assuming no accumulation of 45Ca2+ in extracellular space and constant specific activity for 45Ca 2+ intracellularly. Such fluxes bear no straightforward relationship to the net efflux or influx, except in the special case where one of the unidirectional fluxes far exceeds the other, in which case the dominating unidirectional flux approximates the corresponding net flux. In the case of SIMVI 1, calculation of the unidirectional fluxes for Ca 2+ could be made algebraically from the unidirectional forward and reverse rates of the individual elementary steps in the reaction diagram that were involved in Ca z+ flux, as described in Johnson & Kootsey (1985). In the present study, we decided to use the more straightforward method of simulating the actual isotopic experiment. For SYM56, the relationship between 45Ca2+ states and their unlabeled parents is diagramed in Fig. 7(a) for efflux and 7(b) for influx. Figure 8(a) and (b) shows the corresponding relationships for SYM25. The unidirectional 45Ca2+ fluxes were calculated in the following way, taking 45Ca2+ efflux as an example. The set of differential equations describing the unlabeled exchanger SYM56 were solved for the particular experimental conditions of interest (Cai, Ca,, Nai, Nao, V and T), yielding the fractional amounts of each of the 56 states relative to the total amount of the exchanger. In this way, the fractional amounts of the unlabeled states in Fig. 7(a) (E, ENa, ENa2, ENa3, X, X, e, ECa) could be obtained. With these values, the rate of production of labeled forms of each of the unlabeled states could be calculated as the product of the amount of the unlabeled state and [CaZ+]i. With these rates known, a second set of equations describing the reaction diagram of Fig. 7(a) for 45Ca-labeled states could be solved, yielding the fractional amount of each labeled state relative to the total amount of labeled exchanger. Among these are the fractional amounts of those states from which 45Ca effluxes, namely, ECa, NaECa, Na2ECa, Na3ECa, CaYc,XCa and CaECa. The total efflux is therefore the sum of the products of the amount of each of these states and its associated Ca 2+ dissociation rate constant. 4.4. STATISTICALFITTING OF THE EXPERIMENTAL DATA Statistical fitting of the data was performed using the SCoPFit software created in parallel with SCoP (Kootsey et al., 1986), allowing formal optimization of parameter

458

E. A. J O H N S O N

ET AL.

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