over K in Glial Cells of Bee Retina

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1 Online Supplemental Material for J. Gen. Physiol. Vol. 116 No. 2 p. 125, Marcaggi and Coles A Cl Cotransporter Selective for 4 over K in Glial Cells of Bee Retina Païkan Marcaggi and Jonathan A. Coles SUMMARY A mathematical model is described for calculating fluxes of 4 into and out of a cell from changes in intracellular ph (ph i ) for brief (30-s) applications of ammonium. A kinetic model is then described for a 4 Cl cotransporter with competitive inhibition by K at the 4 sites. Transmembrane Fluxes Associated with Application of Ammonium to a Cell Given the large diffusion coefficient of ammonium in saline solutions ( m 2 s 1 ; Robinson and Stokes, 1959) and the size of the cells (5-m diameter), ammonium is likely to reach equilibrium in the whole cytosol in 10 ms. The processes studied here were on a time scale of seconds, so concentrations can be considered uniform throughout the cytosol. Figure S1. Electrical circuit representation of Fig. 2 C relating membrane fluxes to changes in ph i. Voltages represent concentrations and capacitances represent volumes. A concentration [ 4 o (t) is imposed at node 4 o and gives rise to fluxes between nodes (see text for explanation) Fluxes illustrated in Fig. 2 D ( 4 o 4 i, 4 i 3i, and 3i 3o ) were represented by an electrical circuit diagram (Fig. S1), similar to that of Figure 9 C in Marcaggi et al. (1999), in which voltages, currants, and capacitances correspond to concentrations, fluxes, and volumes. Transport was initiated by imposing on the extracellular node, 4 o, a step potential ([ 4 o ) that reached its full value either instantaneously, or exponentially with 5.5 s (see Figs. S2 and S3, below). Each intracellular node was linked to a capacitance of 1 F (corresponding to a volume of 1 liter) so as to relate charges and voltages (i.e., quantities of molecules and intracellular concentrations). Nodes 4 o and 4 i were linked by F 4, which was either an imposed step function (Figs. 8 and 9) or deduced from the transport scheme of Figs. 10 A and S2 B (Figs. S2, C and D, S3, 10 C, and 11, B and C). Nodes 4 i and 3i were linked by a flux maintaining the equilibrium [ 3 i [H i /[ 4 i and rapid enough to avoid slowing the other fluxes (idem for nodes 4 o and 3o ). Nodes 3i and 3o were linked by a constant conductance P 3 S / V. Finally, node H i was bound to a unit capacitance whose initial potential was [H M. A net addition [H tot of H to the cytosol produces an increase in free H, [H i, given by Eq. 1: δ[ H i δ[ H tot = [ H i ln10 δph i δ[ H tot = [ H i ln10 ( 1 β i ). (1) 1 J. Gen. Physiol. The Rockefeller University Press Volume 116 August

2 Hence, the net flux into node H i was (Eq. 2) F H F reg = [ H i ln10 ( net flux 4 i 3i ) β i ([ H i [ H ) τ reg, (2) where i 12 mm and reg 3 min. The simulations were performed with the program PSpice (MicroSim Corp.). Multistate Model of a 4 -Cl Cotransporter with Competitive Inhibition by K In simple standard models, a full description of the binding and cotransport of two ions (in the present case 4 and Cl - ) requires at least 12 kinetic parameters (Sanders et al., 1984), and a further 6 parameters are needed to account for a competitive inhibition by K on both sides of the membrane (Fig. S2 A). To express our experimental results quantitatively we consider a drastically simplified model. This model then defines apparent binding constants for 4 and K which we can estimate numerically by comparison with the data. The simplification of the model (from Fig. S2 A to S2 B) is equivalent to making the following assumptions. (a) We assume that the binding of K instead of 4 does not affect the kinetics of the transit step of the loaded transporter. It follows that g 1 g 3 and g 2 g 4. (b) We make the arbitrary assumption that the free energy of the transporter and of the transporter plus ion pairs is the same on both sides of the membrane, so that g 1 g 2, g 3 g 4, and k 1 k 2. Since the ion pairs are neutral and the estimated membrane potential is small, it is plausible that this assumption might be approximately true. (c) We assume that the cotransporter exists in equilibrium with transported ligands (Cl, K, and 4 ) so that the three transit steps of the transporter from one side of the membrane to the other side are rate limiting. Each pair of kinetic parameters for the binding of the ligands to the transporter (b j and b j ) can then be replaced by a single affinity constant K j b j /b j. (d) The principle of detailed balance (see Stein, 1986) adds two constraints for the kinetic parameters of the general scheme of Fig. S2 A. We set [K o [K i 0 and [ 4 o [Cl o [ 4 i [Cl i so that 4 and Cl are at equilibrium and the net flux F 4 0. It follows that the cycling flux of transporter states in the negative direction of Fig. S2 A (Xo XClo XCl 4 o XCl 4 i XCli Xi Xo) equals the cycling flux in the positive direction. Thus, b 1 [Xo[Cl o b 3 [XClo[ 4 o g 1 [XCl 4 o b -6 [XCl 4 i b -4 [XCli k 2 [Xi k 1 [Xo b 4 [Xi[Cl i b 6 [XCli[ 4 i g 2 [XCl 4 i b 3 [XCl 4 o b 1 [XClo. Since [ 4 o [Cl o [ 4 i [Cl i, it follows that: g 1 k 2 b 1 b 3 b 4 b 6 = g 2 k 1 b 1 b 3 b 4 b 6. (3) From assumptions a and b, g 1 g 2 g 3 g 4 g and k 1 k 2 k, so Eq. 3 can be written b 1 b b 1 b 3 = b b 4 b 6 b ; i.e., K 1 K 3 = K 4 K 6. (4) The complementary set of conditions with [ 4 o [ 4 i 0 and K and Cl at equilibrium, would lead to: K 1 K 2 = K 4 K 5. (5) Eqs. 4 and 5 are the two constraints for the kinetic parameters of the transporter that follow from the principle of detailed balance. If we then assume that at least one of the outside binding constants is equal to the corresponding inside binding constant (K j K 3j ), it follows that all the binding constants are symmetric and we can write K c K 1 K 4, K i K 2 K 5, K m K 3 K 6, and the scheme of Fig. S2 A reduces to that of B. Calculation of F 4 through the 4 -Cl Cotransporter Model Calculation at steady state (for given binding ion concentrations). By steady state, we mean that the distribution of the transporter molecules between the two faces of the membrane is constant. This cannot be true when the free concentrations of binding ions change, externally because of an applied change and internally as a consequence of the resulting change in F 4. However, if the transit steps are rapid compared with the rates of change of the binding ion concentrations, the rate of change of the distribution of the transporter molecules between the two faces of the membrane can be negligible. The consequences of this approximation will be tested (Fig. S2, C and D). 2 Marcaggi and Coles Online Supplemental Material

3 Figure S2. Kinetic model of the cotransporter. (A) General kinetic scheme of the cotransport of Cl and 4 or K. The unloaded transporter molecule is symbolized by X; o and i indicate the position of the transporter at the external and internal side of the cell membrane, respectively. Binding ions Cl, 4, and K are not shown in this scheme for visual simplification. (B) Simplified kinetic scheme of the cotransport. By means of four assumptions (see text), the kinetic scheme was reduced to a simpler scheme with only five kinetic parameters: three binding constants K c, K m, and K i for Cl, 4, and K irrespective of the side to which the transporter faces, and two kinetic constants k and g for the transit step of the unloaded and loaded transporter. (C) Calculated changes in concentrations of two states of the transporter (XCl 4 o and XCl 4 i) during and after a 30-s application of 2 mm 4. [ 4 o was transiently increased from 0 to 2 mm either instantaneously (dotted trace) or in accordance with the observed time course of solution changes (an exponential with 5.5 s; see Fig. 1 B) (continuous trace). The kinetic scheme of B (with parameters: K c 20 mm, K m 7 mm, K i 20 mm, k 1 s 1, g s 1, n mol m 2 ) was incorporated in the cell model (with parameters: ph i 7.4, i 12 mm, P 3 13 m s 1, reg 3 min) with concentrations of binding ions Cl and K as detailed in the text, and simulations gave the changes in [XCl 4 o (thick traces) and [XCl 4 i (thin traces) for the two patterns of increase of [ 4 o. (D) Calculated ph i changes induced by 2 mm 4 applied for 30 s. All parameters are as in C. Thick continuous and dashed traces show ph i changes (for the two patterns of increase of [ 4 o ) obtained from a simulation with the transport scheme of B included in the cell model; F 4 g[xcl 4 o g[xcl 4 i. Thin trace shows ph i change (for the exponential onset of the change in [ 4 o ) obtained from a simulation with F 4 of Eq. 9 (Table I). 3

4 The calculation was made for the scheme of Fig. S2 B. The affinity constants for 4, K, and Cl are defined as follows. K m K i K c 4 4 [ o [ XClo [ i [ XCli = = [ XCl 4 o [ XCl 4 i ; [ o [ XClo [ i [ XCli = = [ XClKo [ XClKi ; K [ Cl o [ Xo = = [ XClo From the conservation of matter, the sum of the concentrations of the various states of the transporter X remains constant: K [ Cl i [ Xi [ XCli [ Xo + [ XClo + [ XClKo + [ XCl 4 o + [ Xi + [ XCli + [ XClKi + [ XCl 4 i = n K c K m K m K m [ K o [ XCl4 o [ Cl o [ 4 o [ 4 o K i [ 4 o K c K m K m K m [ K i [ XCl4 i = n [ Cl i [ 4 i [ 4 i K i [ 4 i At steady state: δ( [ Xi + [ XCli + [ XClKi +[ XCl 4 i ) = 0, δt and it follows that: g[ XCl 4 o + g[ XClKo + kxo [ = g[ XCl 4 i + g[ XClKi + kxi [ (6) g g K m[ K o kk c K m [ XCl K i [ 4 o Cl 4 o g g K m[ K i kk c K = m [ XCl [ o [ 4 o K i [ 4 i Cl 4 i [ i [ 4 i [ 4 i Cl [ i ( gk i [ 4 o [ Cl o + gk m [ K o [ Cl o + kk i K c K [ XCl 4 i m ) = [ XCl (7) [ 4 o Cl [ o gk i [ 4 i [ Cl i gk m [ K i Cl 4 o. ( + [ i + kk i K c K m ) By combining Eqs. 6 and 7, [XCl 4 o is obtained as a function of n and binding ion concentrations, and then the unidirectional inward flux of 4 (F i ) through the transporter: with F i = g[ XCl 4 o = n ( k g)k c K m [ 4 o [ Cl o [ 4 o [ Cl o [ 4 i [ Cl + i + ( K m K i )[ 4 o [ Cl o [ K i [ Cl i , D (8) D ( 1 g) K c K m K m [ Cl o ( K m K i )[ K o [ Cl o [ 4 o [ Cl = { o }{[ 4 i [ Cl i + ( K m K i )[ K i [ Cl i +( k g)k c K m } + ( 1 g) { K c K m + K m [ Cl i + ( K m K i )[ K i [ Cl i + [ 4 i [ Cl i }{[ 4 o [ Cl o + ( K m K i )[ K o [ Cl o +( k g)k c K m }. The unidirectional outward component of the 4 flux (F o ) is obtained by inverting indices i and o in Eq. 8. The net flux of 4 (F 4 ) is the difference between F i and F o. Development and factorization of D lead to the familiar looking Eq. 9 of Table I, which gives the net flux of 4 through the transporter in a form that makes visible the experimentally accessible transport parameters K, R, R i, R o, R e. Calculation from flux equilibrium. The transporter scheme of Fig. S2 B was represented by an equivalent electrical circuit in which the eight states of the transporter corresponded to nodes Xo, XClo, XClKo, XCl 4 o, Xi, XCli, XClKi, and XCl 4 i, each of which connected to a unit capacitance. Initial potentials at nodes Xo and Xi were n/ 4 Marcaggi and Coles Online Supplemental Material

5 TABLE I Steady State Solution for the 4 (K )-Cl Cotransporter K [ F 4 o Cl ( [ o [ 4 i [ Cl i ) ( K m K i )[ Cl o Cl [ i [ 4 o K + ( [ i [ 4 i [ K o ) 4 = K 2 R KR i [ 4 o [ Cl o KR o [ 4 i [ Cl i R e [ 4 o [ Cl o [ 4 i [ Cl i K k = --K g c K m 1 nr -- [ Cl 1 o [ Cl o [ K o gcl [ 1 i [ K i [ Cl 1 i [ Cl i [ K i gcl [ 1 o [ K o = k + K c K c K i kk c K i K c K c K i kk c K i (9) 1 [ Cl nr i -- 1 i 2 [ Cl i [ K = i k K c K c K i g 1 [ Cl nr o -- 1 o 2 [ Cl o [ K = o k K c K c K i g 2 nr e = -- g This result is for the special case of the kinetic scheme of Fig. S2 B. n is the total number of cotransporters per unit area of membrane. Note that for [K o = [K i = 0 or for K i + one can find again the solution for a kinetic scheme with no competition for the binding site of the substrate (see Stein, 1986). 2 (for [Xo [Xi n). Initial potentials at the other nodes were zero. Each binding step was expressed as a flux rapid enough to avoid slowing the system, such that the binding equilibrium was maintained (for example, [ 4 i [XCli/[XCl 4 i K m ). Nodes XCl 4 o and XCl 4 i were linked by the flux g ([XCl 4 o [XCl 4 i), XClKo and XClKi by g ([XClKo [XClKi), and Xo and Xi by k ([Xo [Xi). This circuit was treated by PSpice simultaneously with the circuit of the cell model (Fig. S1), the two circuits sharing potentials at nodes 4 o and 4 i. F 4 was set to g ([XCl 4 o [XCl 4 i). Testing the Steady State Approximation Initial choice of parameter values. For an initial choice of parameter values for the transporter, K m and K i were made equal to our preferred estimates for K m and K i for the functional transport (K m 7 mm, K i 20 mm). From Fig S3, A C, n k mol s 1 m 2 and n g mol s 1 m 2. We set K c 20 mm. Parameters determined from independent experiments were: P 3 13 m s 1, i 12 mm, baseline ph i ph 7.4, reg 3 min. [K o and [Cl o were the concentrations used in the perfusate, 10 and 222 mm. From the effect of nigericin on ph i, we deduced that a lower estimate of [Cl o /[Cl i was 1.18 (Fig. S3 E); the best fit to the records of ph i was obtained with [Cl i 151 mm (Fig. S3 A). [K i was then calculated from the Gibbs-Donnan relation [K i / [K o [Cl o /[Cl i, which gives [K i 14.7 mm. [Cl o, [Cl i, and [K i were maintained at these values in all further simulations. Is the steady state approximation valid during a 30-s application of 4? To see if the steady state approximation (dxj/dt 0) was satisfactory during brief 30-s 4 applications on cells, we used the cell model as previously described (above), but allowed F 4 to be determined by the transporter model instead of being imposed. [ 4 o was increased from zero for 30 s as illustrated in Fig. S2 C, either instantaneously (dotted trace) or with a diffusion delay (continuous trace). Calculated changes in concentration of the XCl 4 o and XCl 4 i forms of the transporter are shown in Fig. S2 C (bottom). This example shows that for this experimental protocol, the steady state assumption is imperfectly valid. Indeed significant changes in the concentration of each state of the transporter are observed during the whole 30 s of the ammonium application, especially when [ 4 o changes are modeled with a diffusion delay (continuous traces, Fig. S2 C). Nevertheless, Fig. S2 D shows that ammonium-induced ph i changes in a cell model including the transporter scheme of B (thick traces) were not far different from the ammonium-induced ph i changes in a cell in which the transmembrane flux of 4 was set equal to the steady state calculated F 4 (Eq. 9 in Table I) (thin trace). 4 /K Selectivity of the Transporter Model The relation between the [ 4 o, which half saturates inward F 4 in the experimental conditions used, and the binding affinity of the transporter molecule for extracellular 4 is indirect. We used the model transporter of 5

6 Figure S3. Relation between the affinities of the transporter for 4 and K and effects of [ 4 o and [K o on ph i. K c was set to 20 mm. (A) Simulated ph i response to a 10-min application of 2 mm 4 in 10 mm K. n, K m, and K i were set to 1.2 fmol m 2, 100 mm, and 100 mm. To fit the conditions ph i /t ph unit min 1 (between 10 and 30 s after the onset of 4 application) and ph i (after 10 min of 4 application), the remaining parameters of 4 transport were determined as k 1 s 1, g s 1 and [Cl i 151 mm. (B) Couples (K i, g) related to conditions of A. All parameters were set as in A. K i was set to various values and g was calculated to fit the two conditions shown in A. (C) Couples (K m, n) related to conditions of A. All parameters were set as in A. K m was set to various values and n was calculated to fit the condition ph i 6.844, after 10 min of 4 application, shown in A. Plots of such calculated n versus K m (top) and of the resulting ph i /t calculated as above (bottom) are shown for two extreme couples (K i, g) equal to (1 mm, s 1 ) () and (100 mm, s 1 ) (). (D) Example of simulated ph i responses to 30-s applications of 2 mm 4 in 0 and 10 mm K o. In this example, K m 10 mm, K i 20 mm, and other parameters were in accordance with B and C. The mean slopes of ph i responses between 10 and 30 s after the onset of the 4 application were measured as p 1 in 0 K o and p 2 in 10 mm K o. Total ph i changes induced by the 30-s 4 applications were measured as 1 in 0 K o and 2 in 10 mm K o. (E) Plot of pairs (K m, K i ) that fit the conditions p 1 /p () or 1 / (). Other parameters were in accordance with B and C. (F) [ 4 o, which half saturates ph i /t([ 4 o ) (filled symbols) and ph i ([ 4 o ) (open symbols), versus K m of the transporter model. Plots are given for K i 10 mm (triangles) and 20 mm (diamonds). Experimental values that half saturated ph i responses are indicated by dotted lines. Figs. 10 A and S2 B to obtain an idea of how the effect on ph i of increasing [ 4 o and the inhibition by extracellular K can be related to binding affinities for 4 (K m ) and for K (K i ). Simulations were first performed for K c 20 mm, K i 100 mm, K m 100 mm, and n 1.2 fmol m 2 for the case of long applications of 4 at 2 mm on cells with baseline ph i 7.40 and [K o 10 mm. Parameters [Cl i (and as a consequence [K i, see above), k, and g were varied until the two following conditions that correspond to mean values of experimental data were fulfilled: mean ph i /t between 10 and 30 s after the onset of the application of 2 mm ph units 6 Marcaggi and Coles Online Supplemental Material

7 min 1 and ph i after ammonium application lasting at least 10 min. Fig. S3 A shows that these conditions obtained for [Cl i 151 mm, k 1 s 1 and g s 1. Parameters [Cl i and k were then maintained at these values. The procedure was performed with various values of K i and the above two conditions were fulfilled by simultaneously varying g as indicated in Fig. S3 B, with no changes in the other parameters. From now on, values of K i were linked to values of g according to the relation of Fig. S3 B. This procedure was again performed with various values of K m and the second condition (plateau ph i with [ 4 o 2 mm and [K o 10 mm) was fulfilled by simultaneously varying n, as indicated in Fig. S3 C. The bottom plot in Fig. S3 C shows that ph i / t( 4 2 mm in K 10 mm) remains fairly constant for these simultaneous changes of K m and n. Also, since the plots of n versus K m obtained for (K i, g) (1 mm, s 1 ) and (100 mm, s 1 ) were nearly superposable, the couples (K m, n) defined in this way were almost independent of the couples (K i, g) defined in Fig. S3 B. To summarize, any couple (K m, K i ) can fit the two conditions of Fig. S3 A provided that n and g have the corresponding values given by the plots in Fig. S3, B and C. To estimate K i, we considered simulated ph i responses to 30-s applications of 2 mm 4 in 0 and 10 mm external K. Fig. S3 D shows a simulation for (K m, K i ) (10 mm, 20 mm), the responses being characterized by the slopes ph i /t p 1, p 2 and the peak changes ph i 1, 2. K m was then varied and for each value the corresponding K i was calculated to give the ratios p 1 /p and 1 / , which approximate the experimentally observed ones. The resulting pairs of values of K i were plotted versus K m in Fig. S3 E, showing that K i estimated from the inhibition of ph i /t by 10 mm K was in the range mm, and estimated from inhibition of ph i by 10 mm K, mm, irrespective of the value taken for K m in the range mm. On the assumption that these values (10 20 mm) bracket the true K i, the (modeled) [ 4 o that half saturated simulated ph i /t( 4 ) was calculated from ph i responses to 0.5, 1, 2, and 5 mm and plotted versus K m of the model (Fig. S3 F, filled symbols). Similarly, the [ 4 o that half saturated simulated ph i ( 4 ) was estimated from ph i responses to 0.5, 1, 2, 5, 10, and 20 mm and plotted versus K m of the model transporter (Fig. S3 F, open symbols). From comparison with the experimental data of Fig. 7 D, we conclude that K m is in the range 6 8 mm, which is approximately half of K i according to Fig. S3 E. This whole simulation procedure, illustrated by Fig. S3 for P 3 13 m s 1, was also performed for P 3 7 and 19 m s 1 ; the calculated values for K m and K i were little changed and K m /K i remained 0.5. Limits of the Model for the Transport Process In our model of the transporter kinetics, we only considered the case of Cl binding before 4 or K at the outer and inner faces of the membrane. The order of binding of substrates on a K -Cl cotransporter was deduced by Delpire and Lauf (1991), but only on the assumption that the rate limitation occurs in transmembrane transit, so that the ligand-binding reactions are at equilibrium (as for our model; see above). This assumption, as pointed out by Sanders et al. (1984), is physically dubious, and for simple models lacking this assumption, any transport kinetics can be accounted for by appropriate sets of kinetic parameters for the binding steps, irrespective of the order of binding of the substrates. We therefore chose not to investigate different orders of binding of Cl and 4. REFERENCES Delpire, E., and P.K. Lauf Kinetics of Cl-dependent K fluxes in hyposmotically swollen low K sheep erythrocytes. J. Gen. Physiol. 97: Marcaggi, P., D.T. Thwaites, J.W. Deitmer, and J.A. Coles Chloride-dependent transport of 4 into bee retinal glial cells. Eur. J. Neurosci. 11: Robinson, R.A., and R.H. Stokes Electrolyte solutions. 2nd ed. London Butterworths, London, UK. 571 pp. Sanders, D., U.-P. Hansen, D. Gradmann, and C.L. Slayman Generalized kinetic analysis of ion-driven cotransport systems: a unified interpretation of selective ionic effects on Michaelis parameters. J. Membr. Biol. 77: Stein, W.D Transport and diffusion across cell membranes. Academic Press, Inc., Orlando, FL