INTRODUCTION

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The high-affinity, Na⁺-D-glucose cotransport system (SGLT1) has been considered throughout the years as representative of a larger class of membrane transport proteins that use electrochemical gradients for ions to accumulate organic molecules in a variety of prokaryote and eukaryote cells (Schultz and Curran, 1970; Crane, 1977; Semenza et al., 1984; Kimmich, 1990; Wright, 1993). The structural basis underlying the specific and efficient coupling of glucose transport to Na⁺ flux is currently unknown, so most of our understanding of the coupling process per se relies heavily on a number of kinetic studies performed in brush-border membrane vesicles (Hopfer and Groseclose, 1980; Kessler and Semenza, 1983; Moran et al., 1988; Koepsell et al., 1990; Chenu and Berteloot, 1993), intact cells (Restrepo and Kimmich, 1985a,b, 1986; Kimmich and Randles, 1988), and SGLT1 cRNA-injected oocytes (Parent et al., 1992a; Loo et al., 1993; Hazama et al., 1997).

Most contemporary approaches to Na⁺/glucose cotransport kinetics involved the mobile carrier concept, which mainly states that the activator (Na⁺, N) and substrate (glucose, S) binding sites are only accessible from one side of the membrane at a time through isomerization of the unloaded transporter (Schultz and Curran, 1970; Heinz et al., 1972; Jacquez, 1972; Crane, 1977; Stein, 1981; Turner, 1981, 1983, 1985; Sanders et al., 1984; Läuger and Jauch, 1986). In this concept, coupling of the N and S fluxes results from the formation of an N-S carrier complex that undergoes another conformational change, accounting for the differences in compartmentalization of the two molecules during the transport cycle. With the demonstration that SGLT1 couples the transport of two Na⁺ ions to one glucose molecule (Kimmich and Randles, 1980; Moran et al., 1988; Parent et al., 1992a; Chen et al., 1995), and because ordered substrate addition is usually assumed (Kimmich, 1990; Wright, 1993), it turns out that the simplest models of cotransport mostly considered in recent kinetic studies all belong to the family of terreactant systems, which is composed of the three possible eight-state sequential iso ordered ter ter mechanisms, i.e., the so-called S:N:N, N:S:N, and N:N:S models (Restrepo and Kimmich, 1985a). The first possibility seems to be ruled out by those kinetic studies, showing that Na⁺ is mandatory for both glucose cotransport and phlorizin (a specific inhibitor of SGLT1) binding to the carrier (Restrepo and Kimmich, 1985b, 1986; Chenu and Berteloot, 1993; Oulianova and Berteloot, 1996). The second possibility is advocated by the Kimmich group (Kimmich, 1990), and its value as a likely kinetic scheme of Na⁺/glucose cotransport relies on the validity of the assumption previously dismissed by Hopfer and Groseclose (1980) that N and S binding to the carrier occurs under rapid equilibrium conditions. Accordingly, these two kinetic schemes will not be considered any further in the present studies, although we acknowledge the fact that the N:S:N model has been used to satisfactorily describe a number of experimental results (Smith-Maxwell et al., 1990; Bennett and Kimmich, 1996). The last possibility is currently supported by a vast body of literature, mostly published by the Wright group (Wright, 1993) and involving steady-state and presteady-state kinetic studies in SGLT1 cRNA-injected oocytes (Parent et al., 1992a; Loo et al., 1993; Hazama et al., 1997). The characteristic feature of the kinetic scheme proposed by this group is that the expected eight-state N:N:S mechanism of cotransport was reduced to a six-state model in which the binding of the two Na⁺ ions to the carrier is described as a single reaction step (Parent et al., 1992b). However, no direct proof of the two hypotheses originally claimed by Parent et al. (1992b) has ever been provided by the authors to justify such a simplification procedure.

So far, the major deterrent to the consideration of general ter ter mechanisms of cotransport has been the forbiddingly tedious nature of the derivation of steady-state velocity equations therefrom. Indeed, the number of valid King-Altman patterns would increase from 15 to 56 when going from the model of Parent et al. (1992b) to its eight-state counterpart, and a number of simplifications other than the reduction proposed by these authors have been consistently introduced in the past to minimize the complexity of the derivation process. Note that the latter usually involved various assumptions about the rate-limiting steps in the transport cycle (rapid equilibrium assumption) (Turner, 1981, 1983, 1985; Kimmich, 1990), the symmetry of binding events at the two membrane faces (Jacquez, 1972; Kimmich, 1990; Parent et al., 1992b), the symmetry of translocation rate constants (Schultz and Curran, 1970), and the lack of mobility of partially loaded forms of the carrier (Kimmich, 1990). To our knowledge, the effect of these assumptions on the mathematical expression of the rate law characterizing the most general model has never been evaluated.

From a theoretical point of view, it would appear that the derivation of a cotransport model should be as general as possible. However, to decrease the number of significant parameters that can be determined at the experimental level, it is usually recommended to use the simplest model predicting an equivalent rate law (Stein, 1981). Similarly, if the general form of a velocity equation relative to its dependence on a specific substrate concentration has been established experimentally, the rate law predicted by the most general model might prove more complex than necessary to explain the behavior of the real system (more terms than needed would appear in the theoretical velocity equation). For practical purposes, then, a number of variations in the design of a basic model often need to be evaluated, and with complex mechanisms at least, such a systematic approach is necessarily tedious and prone to human errors. It would thus be most advantageous to resort to computer programs to derive velocity equations; however, none of those previously published in the field of enzyme kinetics (Rhoads and Pring, 1968; Hurst, 1969; Cornish-Bowden, 1977; Fromm, 1979; Herries, 1984; Ishikawa et al., 1988) appeared to us to be readily suitable for achieving this goal.

In the present studies, we first present a new computer program that should allow anyone to derive velocity and binding equations for complex kinetic mechanisms involving multireactant systems with or without rapid equilibrium segments. The usefulness of the program is next demonstrated by deriving the rate equation of the general eight-state N:N:S mechanism of Na⁺/glucose cotransport and by testing a number of hypotheses that would justify its reduction to the six-state model of Parent et al. (1992b). It is finally shown that the unique solution to the latter problem involves high positive cooperativity for Na⁺ binding within a rapid equilibrium segment before a slow isomerization step itself, followed by either steady-state or rapid equilibrium addition of glucose, a kinetic mechanism that is best rationalized when assuming a dimeric structure of the SGLT1 protein.

	MATERIALS AND METHODS

Top Abstract Introduction Materials & Methods Results Discussion Appendix References

Theory

General considerations

Calculation of rate equations using the determinant method

(1)

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View larger version (14K): [in this window] [in a new window]   FIGURE 1   Kinetic models of Na+-D-glucose cotransport. (A) Sequential iso ordered ter ter mechanism in which the two Na+ ions bind first to the carrier and mirror symmetry is assumed. (B) Reduced six-state model proposed by Parent et al. (1992b), in which the two Na+ ions add simultaneously to the carrier in a steady-state fashion. Note that the numbering of the different carrier species in B is different from that proposed originally by these authors and has been chosen to make easier a direct comparison with the model in A. In both A and B, Nao and Nai stand for external (OUT) and internal (IN) Na, and So and Si represent external and internal glucose, respectively. Further details are given in the text.

Application to transport models involving rapid equilibrium segments

View larger version (15K): [in this window] [in a new window]   FIGURE 2   Simplified models of Na+-D-glucose cotransport. (A) It is assumed, as proposed by Parent et al. (1992b), that rapid equilibrium binding of the first Na+ ion is followed by steady-state addition of the second Na+ ion. x and y represent the two blocks within the transport cycle (marked in dashed lines), where the carrier species are linked through a rapid equilibrium reaction. (B) Equivalent six-state model reduced according to Cha's rules (Cha, 1968). NX and NY represent the summation of those carrier species that are linked through a rapid equilibrium sequence in the transport cycle. In both A and B, the numbering of the carrier species is similar to that indicated in the legend to Fig. 1. Further details are given in the text.

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Computerization of the determinant method

The main features of the program are presented in Fig. 3 as a flowchart. The program was written in Mathematica language, which is well suited for mathematical applications like linear algebra, analytical calculations, and algebraic manipulations (Wolfram, 1993). Moreover, in contrast to traditional programming languages such as Fortran or BASIC, which handle only numerical computations, Mathematica also performs symbolic and graphical computations (Wolfram, 1993).

View larger version (19K): [in this window] [in a new window]   FIGURE 3   Flowchart of the computer program developed in the present studies

The data input method is illustrated in the Appendix, using the kinetic mechanism of cotransport shown in Fig. 2 A. The process is interactive, includes several tests to avoid human errors, and can be checked as a whole from the summary table presented at the end of the input session. The latter can be used further as a guide to performing necessary corrections in the current model or to introduce modifications in a preedited model file. Note that the program self-establishes, for any particular model, its characteristic matrix Eq. 10 and, if required by the model, the algebraic expression of the dissociation constants involved in the rapid equilibrium reactions according to their definitions, as in Eqs. 13 and 14.

The program output may then provide the user with the algebraic expressions of (see Appendix) 1) the fractional concentration factors f_i calculated according to their definitions, as in Eqs. 17-20, if any; 2) the velocity or binding equation relative to the release of a particular product or fixation to the transporter of any molecule included in the basic model (substrate, inhibitor, etc.); and 3) the steady-state distribution equations relevant to all carrier species involved in a particular model. Note that the velocity/binding equation is displayed in one run under its generic form, which is the most useful one for experimental applications, and that the algebraic expressions of the constant and substrate coefficient terms can be listed upon request.

Implementation

The program was written for a text-based interface configuration of Mathematica and was implemented on the mainframe computer at the University of Montreal (UNIX system). It has since been run with similar efficiency on a Pentium PC Pro (200 MHz, 32 Mbytes RAM) with a notebook configuration of Mathematica. In its present form, the program has been devised without any limitation to the number of either carrier species or elementary reactions involved, but these may be dictated by the memory space available on specific computers.

	RESULTS

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Solution of the eight-state model of cotransport (Fig. 1 A)

For the purpose of the present studies, zero-trans initial rates of transport only will be considered, because such velocity equations have been used by Parent et al. (1992b) to analyze their experimental data. Relative to glucose (v_i^S) and Na⁺ (v_i^Na) transport, their generic forms resulting from computer calculation are shown in Eqs. 22 and 23:

(22)

(23)

respectively, with constant and substrate coefficient terms (macro constants) as listed in Table 1. Although presented differently to emphasize the fact that S (Eq. 22) or Na⁺ (Eq. 23) can be taken as the variable substrate, the denominator expressions of Eqs. 22-23 are, in fact, identical, as conditioned by the conservation Eq. 9. However, their numerator expressions are quite different and reflect the fact that glucose and Na⁺ are released through alternative pathways on the trans side. Note, then, that the use of Eq. 22 assumes that glucose transport is measured by a radio tracer technique, whereas Eq. 23 may apply to either ²²Na⁺ uptake or the Na⁺ current flowing through the Na⁺ and glucose transport pathways. It follows from these considerations that the measure of the v_i^Na/v_i^S ratio (Lee et al., 1994) may not be the best method for determining the coupling stoichiometry of cotransport, because noninteger numbers are generally expected from this approach. Indeed, it can easily be verified from Table 1 that a true coupling stoichiometry of 2.0 may only be observed in the total absence of Na⁺ leak pathways (i.e., when k₃₈ = k₄₇ = k₇₄ = k₈₃ = 0), in which case a₁ = a₂ = a₃ = 0 and a₄ = 2a₀. Note, however, that the introduction of the latter modifications into the basic model does not affect the general form of the denominator expression of Eqs. 22-23, i.e., none of the constant or substrate coefficient terms can be eliminated by this assumption.

Solution of the six-state model of Parent et al. (1992b) (Fig. 1 B)

In our studies, the numbering of the n carrier species was chosen in such a way that N₁ and N₂ always represent the unloaded forms of the transporter independently of the n value. Moreover, the notation used in Fig. 1 B emphasizes the similarities and the differences between the six- and eight-state models. The initial velocity equations were thus recalculated to fit this new formalism as compared to the original work of Parent et al. (1992b); their generic forms are given by

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for glucose and Na⁺ transport, respectively. The algebraic expressions of the different macro constants are listed in Table 2 and can be compared directly to their counterparts lacking the P indices in Table 1.

View this table: [in this window] [in a new window]   TABLE 2   Derivation of the six-state model of cotransport (Parent et al., 1992b)

A comparison of Eqs. 22-23 with Eqs. 24-25 clearly indicates that the absence of (Na_o) and (Na_o)(S_o) coefficient terms in the denominator expressions of the latter is a major difference between the characteristic velocity equations predicted by the two models shown in Fig. 1, A and B, respectively. Moreover, whereas the numerator expression of v^S_i is similar in form for the two models, that for v_i^Na also lacks the (Na_o) and (Na_o)(S_o) coefficient terms when derived for the six-state model. Because the macro constants a₁ and a₂ in the numerator of Eq. 23 account for a Na⁺ leak pathway through the one Na⁺-loaded carrier complex in the eight-state model (N₃ to N₈ transitions in Fig. 1 A), it can be concluded that an implicit hypothesis of the six-state model of Parent et al. (1992b) is that such a leak does not exist. Indeed, setting k₃₈ = k₈₃ = 0 cancels out the (Na_o) and (Na_o)(S_o) coefficient terms in the numerator of Eq. 23 and emphasizes the 2 factor in the algebraic expressions of the macro constants a₃ and a₄ (Table 1).

Note that the macro constants a_3P and a_4P in Eq. 25 characterize the Na⁺ fluxes through the leak and cotransport pathways, respectively. It is readily apparent, then, that these two fluxes will be equal, provided that Eq. 26 is satisfied:

(26)

The algebraic expression on the right side of Eq. 26 is equivalent, when accounting for the differences in notations, to that previously derived through a different approach by Chen et al. (1995) for the kinetic constant K_c, which was defined as the extracellular sugar concentration giving equal Na⁺ currents through the inward Na⁺ leak and the coupled Na⁺-sugar influx.

The steady-state current flowing through the carrier under zero-trans conditions (I) can be determined from Eq. 25 and is given by Eq. 27:

(27)

in which F is the Faraday constant, and the right side of the last equality corresponds to the expression previously derived by Parent et al. (1992b). Taking into account the differences in formalism, it was verified that the algebraic expressions of the macro constants relative to v_i^Na as defined by Eqs. 25 and 27 satisfy the equalities shown in Eq. 28:

(28)

Still, it should be noted that the 2 factor showing up in the complete form of Eq. 25 (see Table 1) results from our definition of the initial rate, given by

(29)

which accounts for the inside release of two Na⁺ ions. In contrast, it clearly appears that the concept that led Parent et al. (1992b) to the correct Eq. 27 is ill defined, because, as shown by

(30)

(adapted from the original publication to fit the notations shown in Fig. 1 B), the 2 coefficient results from the assumption that the valence of the ion-binding site on the empty carrier (z_c) is equal to 2 (Parent et al., 1992b). Indeed, because the definition of the steady state assumes that there is no net flow of carrier forms from one side of the membrane to the other, a corollary implication is that there should be no net flow through the membrane of those charges that might be involved at the level of the different carrier species. Accordingly, steady-state currents can only measure the charges associated with ion fluxes. This fact has been correctly appreciated in similar studies by Chen et al. (1995), in which the steady-state current was defined as

(31)

(adapted from the original publication to fit the notations shown in Fig. 1 B), where n accounts for the 2:1 stoichiometry of SGLT1. Still, a word of caution should be given regarding the definition of the steady-state velocity equation relative to the free carrier recycling step (N₁ to N₂ transitions) as in Eqs. 30 and 31, rather than to the Na-releasing step as in Eq. 29. Indeed, applying the former concept to the eight-state model (Fig. 1 A) failed to give the correct steady-state equation in the presence of an extra Na⁺ leak pathway represented by the N₃ to N₈ transitions.

First attempts to reduce the general eight-state model of cotransport to the six-states model of Parent et al. (1992b)

Our first attempts to justify a reduction of the general eight-state model of cotransport (Fig. 1 A) to the six-state model shown in Fig. 1 B aimed at testing the two hypotheses proposed by Parent et al. (1992b). The first hypothesis, which assumed that there are two equivalent binding sites for Na⁺ ions on the carrier protein, can easily be introduced into the model of Fig. 1 A by setting k₂₃ = k₃₄, k₄₃ = k₃₂, and k₇₈ = k₈₁ (note that we also assumed that k₃₈ = k₈₃ = 0 as implicit in the model of Parent et al. (1992b); see above). It can easily be verified from Table 1 that these equalities did not allow us to cancel any of the (Na_o) and (Na_o)(S_o) coefficient terms in the denominator of Eqs. 22-23. This result is not unexpected, however, when considering the rule stating that consecutive steps are not additive (Volkenstein and Goldstein, 1966). The second hypothesis, which assumed that binding of the first Na⁺ ion to the carrier is in rapid equilibrium and that binding of the second Na⁺ ion to the carrier is rate limiting, can be introduced into the model of Fig. 1 A as shown in Fig. 2. The computer analysis of this model (see Appendix) led to a v_i^Na equation similar in form to Eq. 23. with macro constants as listed in Table 3 (note that a₁ = a₂ = 0, because of the additional constraint that k₃₈ = k₈₃ = 0 in these calculations). Accordingly, the two hypotheses proposed by Parent et al. (1992b) must be rejected.

Our further attempts to reduce the eight-state model of cotransport involved the analysis of models lacking the N₃-N₈ transition in Fig. 1 A and differing in the number and position of the rapid equilibrium steps that can be introduced into the transport cycle. Their characteristic velocity equations were established using our computer program, and the results showed that v_i^Na was similar in form to Eq. 23 (with a₁ = a₂ = 0), with a number of modifications in some of the substrate coefficient terms in the denominator as listed in Table 4. Note that 1) it does not matter to the general form of Eq. 23 whether S binds to the carrier in a steady-state or rapid equilibrium reaction, a conclusion that was verified to apply to the model of Parent et al. (1992b), so that Eqs. 23 and 25 do not represent a unique solution to the full steady-state models shown in Fig. 1, A and 1 B, respectively; 2) all models in which c₀ = 0, including those leading to the disappearance of the (Na_o)(S_o) coefficient term c₁ in the denominator of Eq. 23, must be rejected because the rate law characterizing the model of Parent et al. (1992b) contains a finite c_0P constant (see Eq. 25); and 3) in no case was it possible to cancel out both of the macro constants b₁ and c₁ defining, respectively, the (Na_o) and (Na_o)(S_o) coefficient terms in the denominator of Eq. 23. The latter point is not unexpected, because the macro constant b₁ expresses the fact that one of the two Na⁺ ions must precede further Na⁺ and glucose binding to the transporter, as clearly indicated in Table 4 from the analysis of the simplified model, in which all reactants add to the transporter under rapid equilibrium conditions.

Conditions of application of the model of Parent et al. (1992b)

The above studies demonstrate that there is no kinetic basis so far that could justify the reduction of the general eight-state model of cotransport (Fig. 1 A) to the six-state model shown in Fig. 1 B. It can be observed, however, that Eq. 25 is equivalent to the Hill equation,

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in which the parameters V_maxP^Na and K_mP^Na would represent the apparent V_max and K_m of Na⁺ transport estimated at a fixed concentration of glucose (their algebraic expressions can be easily determined from Eq. 25 after dividing both its numerator and denominator by the coefficient of the (Na_o)² term in the denominator) and the Hill number n_H = 2. In agreement with this view and, thus, attesting at the experimental level to the validity of Eq. 32, Parent et al. (1992a) reported n_H values of 1.9-2.1 for Na⁺ activation of the steady-state currents through SGLT1. One may then question the conditions under which Eq. 23 (with a₁ = a₂ = 0, see justifications above) could be approximated by Eq. 32.

To answer this question and to allow easier comparison of Eqs. 32 and 23, it is most convenient to analyze the latter under its equivalent form:

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which is composed entirely of kinetic constants, defined as

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Note that Eqs. 35 and 36 would characterize the apparent affinities of Na⁺ binding to the first and second carrier sites, respectively. Because the denominator of Eq. 33 can be factorized under the two forms shown in Eq. 37,

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it will degenerate to the denominator form of Eq. 32 when K_m2^Na  (Na_o)  K_m1^Na. Accordingly, the model of Parent et al. (1992b) may approximate the general eight-state mechanism of cotransport over a restricted range of Na⁺ concentrations, provided that the apparent affinity for binding of the second Na⁺ ion is much higher than that of the first.

Alternatively, it is possible to estimate the apparent n_H value that one may expect from a Hill plot analysis of Eq. 33. Because n_H is given in this approach by the slope of the Ln[v/(V_max  v)] versus Ln(S) plot at v = V_max/2 (Segel, 1975), its value can be evaluated from

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which simply translates this fact into mathematical terms. Equation 38 can be transformed to its equivalent form,

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after using the mathematical rules governing the derivation of logarithmic functions. When applied to Eq. 33, formal development of the terms in brackets in Eq. 39 and further rearrangements lead to

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from which it is readily apparent that 1  n_H  2 and n_H = 2, provided that (Na_o)_0.5  K_m1^Na. The (Na_o)_0.5 expression can be found by solving Eq. 33 for (Na)_o after setting v_i^Na = V_max^Na/2, and the result is given by

(41)

so that n_H = 2, provided that 2K_m2^Na  K_m1^Na. Accordingly, then, the model of Parent et al. (1992b) may also approximate the general eight-state mechanism of cotransport in the case of very strong positive cooperativity for Na⁺ binding to the transporter.

High positive cooperativity for Na⁺ binding as a likely physical basis of the kinetic features of the six-state model of Parent et al. (1992b)

It should be acknowledged that the concept of positive cooperativity for Na⁺ binding was coined by Parent et al. (1992b) in their original paper. However, it was neither demonstrated as such nor discussed further by these authors. Indeed, its formal development into a realistic kinetic scheme may have to address first a number of issues that have been overlooked so far, in part to keep the analysis of the general eight-state mechanism of cotransport as simple as possible, but also because most of these questions are not readily apparent from the model of Parent et al. (1992b). First, with a transporter known to possess two Na⁺ binding sites, it is very unlikely to have a purely ordered mechanism of Na⁺ binding unless there is extensive asymmetry in the configuration of the two sites. The demonstration by Levitzki (1978) that the latter situation is able to account for noncooperativity and negative cooperativity, but never for positive cooperativity, would suggest that a random addition of the two Na⁺ ions represents the most relevant mechanism to be considered in the case of Na⁺-D-glucose cotransport. Second, given the assumption of steady-state reactions for Na⁺ binding in a random mechanism, our computer calculations showed that the resulting velocity equation contains (Na_o)³ coefficient terms that do not exist in Eqs. 23 and 25. There is, thus, a rational basis for assuming that the random addition of the two Na⁺ ions should occur within a rapid equilibrium segment of the transport cycle. Last, because the concept of a purely kinetic nature of positive cooperativity would appear to be incompatible with the rapid equilibrium assumption above (Segel, 1975) and with the observation that equilibrium binding curves of the nontransported inhibitor phlorizin are sigmoidal relative to Na⁺ concentrations (Restrepo and Kimmich, 1986; Moran et al., 1988), we assume in the following that the Na⁺ sites are interactive, i.e., the first occupancy of one of the two sites modifies the apparent affinity for Na⁺ binding to the unoccupied site.

According to these considerations, the binding sequence of the two Na⁺ ions can be depicted as shown in Fig. 4 A and corresponds to a block x in the transport cycle, verifying Eq. 42:

(42)

in which K_Na, the intrinsic dissociation constant for binding of the first Na⁺ ion, is modified by a factor  accounting for the increased affinity of Na⁺ binding to the second site ( < 1 for positive cooperativity). The fractional concentrations of the N_i carrier forms within the block x can thus be expressed by

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from which it can be concluded that f₃  f₂ and f₄, if 2K_Na  (Na_o)  K_Na/2. Thus f₃  0 when   1/4, and this situation is equivalent to the simultaneous addition of two Na⁺ ions with an apparent dissociation constant of K_Na². Note that, under these conditions, the low concentration of the N₃ carrier species would also justify the approximation that k₃₈ = 0, as implicitly assumed by Parent et al. (1992b) (see above). Still, as shown in Table 4, a general eight-state mechanism of cotransport in which the two Na⁺ binding steps occur under rapid equilibrium conditions cannot pretend to mimic the behavior of the model of Parent et al. (1992b). Nonetheless, if we further assume the existence of a slow isomerization step between the Na⁺ and glucose binding sequences, as depicted in Fig. 4 B, the characteristic velocity equation calculated by our computer program is similar in form to Eq. 25, with algebraic expressions of the macro constants as given in Table 5.

View larger version (13K): [in this window] [in a new window]   FIGURE 4   Positive cooperativity in Na+ binding to the cotransport protein. (A) It is assumed that the conformations of the two Na+ sites present on the free carrier facing toward the outside medium are equivalent (the free carrier is symmetrical), so that binding of the first Na+ ion may occur at either site with similar affinity (dissociation constant KNa). It is also assumed that the first Na+ ion to bind induces a conformational change that results in increased affinity of Na+ binding to the second site (dissociation constant KNa, where   1 for high positive cooperativity). (B) Proposed eight-state model of Na+-D-glucose cotransport that includes the concept of high positive cooperativity developed in A. Note that a slow isomerization step has been included between the Na+ and glucose-binding sequences for reasons discussed in the text. (C) Equivalent six-state model of the mechanism shown in B after reduction according to Cha's rules (Cha, 1968). NX and NY represent the summation of those carrier species that are linked through a rapid equilibrium sequence in the transport cycle. In both A and B, the numbering of the carrier species is similar to that shown in Fig. 1. In C, however, it is similar to that proposed originally by Parent et al. (1992b), to underscore the similarity of the two models (k* corresponds to their definition in the original publication by these authors, whereas k is equivalent to those shown in B). Further details are given in the text.

	DISCUSSION

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