 Methodology article
 Open Access
 Published:
A novel delta current method for transport stoichiometry estimation
BMC Biophysics volume 7, Article number: 14 (2014)
Abstract
Background
The ion transport stoichiometry (q) of electrogenic transporters is an important determinant of their function. q can be determined by the reversal potential (E_{rev}) if the transporter under study is the only electrogenic transport mechanism or a specific inhibitor is available. An alternative approach is to calculate delta reversal potential (ΔE_{rev}) by altering the concentrations of the transported substrates. This approach is based on the hypothesis that the contributions of other channels and transporters on the membrane to E_{rev} are additive. However, E_{rev} is a complicated function of the sum of different conductances rather than being additive.
Results
We propose a new delta current (ΔI) method based on a simplified model for electrogenic secondary active transport by Heinz (Electrical Potentials in Biological Membrane Transport, 1981). ΔI is the difference between two currents obtained from altering the external concentration of a transported substrate thereby eliminating other currents without the need for a specific inhibitor. q is determined by the ratio of ΔI at two different membrane voltages (V_{1} and V_{2}) where q = 2RT/(F(V_{2} –V_{1}))ln(ΔI_{2}/ΔI_{1}) + 1. We tested this ΔI methodology in HEK293 cells expressing the elctrogenic SLC4 sodium bicarbonate cotransporters NBCe2C and NBCe1A, the results were consistent with those obtained with the E_{rev} inhibitor method. Furthermore, using computational simulations, we compared the estimates of q with the ΔE_{rev} and ΔI methods. The results showed that the ΔE_{rev} method introduces significant error when other channels or electrogenic transporters are present on the membrane and that the ΔI equation accurately calculates the stoichiometric ratio.
Conclusions
We developed a ΔI method for estimating transport stoichiometry of electrogenic transporters based on the Heinz model. This model reduces to the conventional reversal potential method when the transporter under study is the only electrogenic transport process in the membrane. When there are other electrogenic transport pathways, ΔI method eliminates their contribution in estimating q. Computational simulations demonstrated that the ΔE_{rev} method introduces significant error when other channels or electrogenic transporters are present and that the ΔI equation accurately calculates the stoichiometric ratio. This new ΔI method can be readily extended to the analysis of other electrogenic transporters in other tissues.
Background
Based on their electrical properties, membrane protein transporters are classified as being either electrogenic (transport a net charge) or electroneutral [1][3]. Which of these categories a given transporter belongs to is dependent on its substrate (or ion) coupling ratio; its transport stoichiometry represented by the symbol q. Electrogenic transporters are sensitive to both the electrical and chemical gradients of the ions that are being transported across a membrane. Unlike electroneutral transporters, electrogenic transporters can utilize the membrane potential of a cell or organelle membrane to drive substrates or ions against their chemical gradients. For a given electrochemical gradient, the transport stoichiometry is therefore an important independent determinant of both the magnitude and direction of substrate or ion flux through a membrane transport protein. The simplest stoichiometry for an electrogenic transporter is 1:1 as in the case of the sodiumcoupled glucose transporter SGLT2 [4]. In many instances more complex stoichiometries have been reported [4],[5]. Furthermore, certain transporters have variable stoichiometry ratios [6][10].
The most intuitively straightforward approach for measuring the stoichiometry of a transporter is to measure the flux of each transported species either directly [11] or indirectly [12]. In many instances, technical difficulties or sensitivity/specificity considerations preclude interpretable flux measurements from being acquired. Rather than measuring the actual substrate fluxes, a widely used approach is to measure the steady state currentvoltage (IV) properties of the transporter. In this approach, one determines the reversal potential (E_{rev}), and estimates q as for example in the case of an electrogenic sodium coupled bicarbonate transporter [1] as follows:
where intracellular concentrations of Na^{+} ([Na^{+}]_{i}) and HCO_{3} ^{−} ([HCO_{3} ^{−}]_{i}) as well as extracellular concentrations of Na^{+} ([Na^{+}]_{o}) and HCO_{3} ^{−} ([HCO_{3} ^{−}]_{o}) are known and E_{NBC} is the reversal potential of the transporter. F, R and T are Faraday’s constant, gas constant and absolute temperature respectively. RT/F = 25.69 at 25°C [13].
If the electrogenic transporter under consideration is the only transport mechanism in the membrane, q estimated by solving Eq. 1 is accurate. In most cells or expression systems, there are other channels or electrogenic transporters in the membrane, reversal potential method requires the use of a specific inhibitor to differentiate the transport process of interest from other transport pathways. Subtracting the IV curve in the presence of the inhibitor from the IV curve without inhibitor, one obtains the E_{rev} of the transportermediated current. Therefore, the relationship of Eq. 1 still holds.
Given that inhibitors are not always as specific as one would prefer, or in circumstances where a specific inhibitor is unavailable, an alternative approach has been to measure the change in zerocurrent membrane potential (V_{I=0}, the voltage of the IV curve measured at I = 0), by altering the chemical gradient(s) of the transported species [15][16]. Then ΔE_{rev} is
There are some variations of the ΔE_{rev} approach such as estimating q by determining the slope of V_{I=0} vs. ion or substrate concentrations [2]. In this report, we show that ΔE_{rev} approach is correct only when the transport current under study is the only current in the membrane or in other words, currents mediated by other channels, electrogenic transporters, and leak current are negligible. When the currents mediated by other channels/transporters are not negligible, the implicit assumption underlying the ΔE_{rev} approach and its variations is that the reversal potentials due to other channels and transporters are additive to the E_{rev} of the transporter under study, therefore they can be eliminated by subtraction. However, the assumption that E_{rev} is additive is not valid since the effect of multiple channels/electrogenic transporters on ΔE_{rev} is a complicated function of the concentrations of ions and substrates involved, as well as the conductance and transport rate of those pathways [17],[18].
To address these issues, we have developed a new approach named the “delta current (ΔI) method”. The utility of the ΔI approach is demonstrated using the electrogenic sodium bicarbonate cotransporters NBCe2C and NBCe1A [14],[19][21] expressed in HEK293 cells. In vivo, NBCe2C is expressed in choroid plexus epithelial cells and other tissues. NBCe1A is expressed in the mammalian kidney proximal tubule and the eye. This method has several advantages: 1) The equation does not suffer from the aforementioned errors in the ΔE_{rev} method due to other channels and functional electrogenic transporters; 2) Like the ΔE_{rev} method, the measurement protocol does not require a specific inhibitor. In addition, by computational simulations, we show the advantage of the ΔI method in calculating the stoichiometry ratio of an electrogenic transporter, and demonstrate that the ΔE_{rev} method can introduce significant errors in estimating q.
Methods
Expression of NBCe2C and NBCe1A in HEK293 cells
The SLC4 human NBCe2C and NBCe1A proteins were expressed in HEK293 cells as follows. Fulllength human cDNA for each transporter was cloned into a pMSCVIRESEGFP (Clontech, Mountain View, CA) which expresses the transporters under a CMV promoter and also expresses EGFP as a separate protein under an internal ribosome entry site. The cDNA sequence of each of the constructs was verified by DNA sequencing. Use of human material and cell line are approved by UCLA Institutional Biosafety Committee (IBC#111.13.0r).
Electrophysiological recordings
Cells expressing each transporter were cultured in DMEM media with 5% FBS/5% CO_{2} and 37°C. The cells were transferred to 35 mm tissue culture (Bioptechs, Butler PA) inserts that were placed on the microscope stage for patchclamp recording. The cells were continually superfused with bath solution (~2 ml/min) during the experiments. All experiments were performed in room temperature (22 ± 1°C). HEK293 cells were wholecell patchclamped with the aid of fluorescent optics (Axioskop2, Carl Zeiss, Göttingen, Germany). Patch pipettes were pulled from thick wall (0.32 mm) borosilicate glass with tip size 1  1.5 μm (resistance: 46.5 MΩ). The patch pipette filling solution and bath solution components are listed in Table 1. All solutions were pH 7.4 that were confirmed with pH meter measurements in conditions throughout the studies. To ensure stable electrode potentials during wholecell patchclamp recordings, a microagar salt bridge of 2 M KCl was built in the electrode holder that formed an electrical connection between the pipette solution and the Ag/AgCl wire connected to the headstage of a patchclamp amplifier [22]. Intracellular signals were amplified and low passfiltered at 400 Hz with a patchclamp amplifier (MultiClamp 700B, Molecular Devices Co., Sunnyvale, CA). Whole cell capacitance and series resistance were determined with the auto wholecell capacitance and series resistance compensation. The series resistance was usually compensated 80% (both prediction and correction). Junction potentials generated by different pairs of patch pipette solutions and bath solutions were determined with the junction potential calculator in software Clampex 10 (Molecular Devices Co., Sunnyvale, CA) and reported potential values were corrected for junction potentials. The inhibitor 4,4′Diisothiocyanatostilbene2,2′disulfonic acid disodium salt (DIDS; SIGMAAldrich Co., St. Louis, MO.) was used to block NBCe2C and NBCe1A function.
Data analysis
Signals from intracellular recordings were digitized at 2 KHz sampling frequency with the Digidata 1440A and software Clampex 10 (Molecular Devices Co., CA, USA). The signals were saved as data files for further analyses offline. Data are expressed as mean ± SE. Paired ttest was used for determining statistical significance. p ≤ 0.05 was taken as the criterion for significance.
Results
Estimation of NBCe2C transport stoichiometry with the conventional reversal potential method
The light microscopic image of cultured HEK293 cells and corresponding fluorescent image of the same field is shown in Figure 1a and b respectively. Bright fluorescent cells were EGFP positive and thus were NBCe2C expressing cells as well. We voltageclamped EGFP positive cells at a holding voltage 60 mV and applied a series of 400 ms pulses from 95 to +45 with increment of 10 mV. The current responses to the series of pulses in preHCO_{3} ^{−} (0 HCO_{3} ^{−}) conditions were background current due to endogenous channels in HEK293 cells (Figure 2a left panel). We established an IV curve of steady state current. Figure 2b shows the mean IV curves from 8 cells. The steady state current at +45 mV was 51.8 ± 18.0 pA (mean ± SE, n = 8). Bath application of a solution containing 25 mM HCO_{3} ^{−} (Table 1, bath solution B) induced a voltagedependent current (Figure 2a central panel). The mean IV curve in the presence of HCO_{3} ^{−} is shown in Figure 2b. The steady state current at voltage +45 mV was 133.5 ± 25.5 pA (p = 0.01, paired ttest vs preHCO_{3} ^{−}). The HCO_{3} ^{−}induced current was obtained by subtracting the current traces in the absence of HCO_{3} ^{−} from the current traces in its presence. Figure 2c shows the mean IV curve of HCO_{3} ^{−} induced current. The mean HCO_{3} ^{−}induced current at voltage +45 mV was 81.7 ± 23.3 pA (n = 8). The current was greatly reduced after washing with the control bath solution (Figure 2a right panel). As a separate control, we tested whether the application of HCO_{3} ^{−} containing solution induced any current in EGFP negative cells. As shown in Figure 2d, there is no significant HCO_{3} ^{−}induced current detected in these cells (n = 4). These results indicate that functional NBCe2C is expressed in EGFP labeled HEK293 cells and that NBCe2C transports HCO_{3} ^{−} electrogenically.
To estimate the NBCe2C HCO_{3} ^{−} to Na^{+} transport stoichiometry q, the conventional method of measuring the reversal potential with the inhibitor DIDS was used initially. At known intracellular and extracellular concentrations of Na^{+} and HCO_{3} ^{−}, q could be estimated with Eq. 1.
In this study, HEK293 cells expressing NBCe2C were wholecell patchclamped at 60 mV. V_{I=0} was measured in two independent experiments where [HCO_{3} ^{−}]_{i} and [HCO_{3} ^{−}]_{o} were equal (25 mM), therefore E_{NBC} depended only on [Na^{+}]_{i}/[Na^{+}]_{o}. For every cell recorded, we waited at least 10 min from establishment of wholecell patchclamp to ensure that [Na^{+}]_{i} and [HCO_{3} ^{−}]_{i} were equal to the concentrations of Na^{+} and HCO_{3} ^{−} respectively in the patch pipette solution by diffusion before beginning IV measurement. Current responses to a series of voltage pulses were recorded to establish IV relationship in the absence and presence of DIDS (0.5 mM, Figure 3a). In the first experiment, using [Na^{+}]_{i}/[Na^{+}]_{o} = 40/80 mM (Patch solution d/bath solution C in Table 1), IV curve of steadystate NBCe2C transport current (DIDS sensitive current) was obtained by subtraction of currents in the presence of DIDS from control current (preDIDS). V_{I=0} = 22.3 ± 2.4 mV (n = 3) was obtained (Figure 3a,b and d). To show the mean and variability among cells, this V_{I=0} value was averaged from the V_{I=0} of individual sample cells. Note that this mean V_{I=0} value is very close to the V_{I=0} points where the average DIDSsensitive IV curve crosses the xaxis in (Figure 3b). In the second experiment using [Na^{+}]_{i}/[Na^{+}]_{o} = 25/135 mM (Patch solution c/bath solution B in Table 1), we got V_{I=0} = 43.9 ± 3.5 mV (n = 5, Figure 3c and d). The two V_{I=0} values are close to the calculated E_{NBC} values of 17.8 and 43.3 mV (Eq. 1), respectively, assuming q = 2 (dash lines) while significantly distinct from the calculated values assuming q = 3 (dash lines, Figure 3d). The results indicate that the transport stoichiometry ratio of NBCe2C is 2 HCO_{3} ^{−}: 1 Na^{+} or (1 CO_{3} ^{2−}: 1 Na^{+}) in HEK293 cells.
A novel delta current method for estimation of transport stoichiometry
Based on a simplified model for electrogenic secondary active transport [23] (as originally applied to the Na^{+}/Ca^{2+} transporter), in the case of an electrogenic NBC transporter, the Na^{+}HCO_{3} ^{−} flux (J_{c}) is shown in Eq. 2. Although we limit our evidence for the validity of our method to electrogenic NBC transporters, the approach is applicable to other electrogenic transporters.
where K_{c} is an involved function of mobility and concentrations of free and loaded carrier [23] (also refer to [24]). z_{Na} is the valence of Na^{+} and ν_{Na} is the stoichiometry of Na^{+}. ν_{HCO3} is the stoichiometry of HCO_{3} ^{−}. V is the membrane potential. The total membrane current is:
Where $\sum _{j}{I}_{j}$ is the sum of all other currents mediated by various channels and electrogenic transporters including leak current on the membrane. $\sum _{j}{I}_{j}$ can be a nonlinear function of V while a general assumption is that it is independent of NBC transport current.
If we change the Na^{+} concentration outside the cell from [Na^{+}]_{o1} to [Na^{+}]_{o2}, the whole cell current would change from I_{M1} to I_{M2}. We assume that K_{c} does not vary with [Na^{+}]_{o} within a range far from saturation. We also assume that the sum of other currents $\sum _{j}{I}_{j}$ is a function of V while the function is unchanged when [Na^{+}]_{o} changes (see Discussion). Therefore the delta current is
$\sum _{j}{I}_{j}$ is completely eliminated. For simplicity, we take ν_{Na} = 1 and q = ν_{HCO3}/ν_{Na}.
Now we consider at two different voltage points V_{1} and V_{2}, we have two ΔI_{M} values, ΔI_{V1} and ΔI_{V2}. We take the ratio of them,
ΔI_{V1} and ΔI_{V2} can be measured in electrophysiological experiments, therefore, there is only one unknown q. q can be expressed as
In practical situations, to minimize the effect of the possible voltage dependence of K_{c} on the measurement of ΔI_{M} and estimation of q, we take [Na^{+}]_{o1} = [Na^{+}]_{i} and [HCO_{3} ^{−}]_{o} = [HCO_{3} ^{−}]_{i}, where
Therefore, at V = 0, the delta current ΔI_{V1=0} is the pure NBC transport current at [Na^{+}]_{o2}.
q is as simple as
In the following applications, to minimize the effects of possible K_{c} voltage dependence, we also take a V_{2} value close to 0 (e.g. ± 10 to 15 mV). Therefore the calculation involves only experimental measurements of currents close to equilibrium conditions.
Transport stoichiometry of NBCe2C estimated with the delta current method
Under the conditions that [Na^{+}]_{i} = [Na^{+}]_{o} = 10 mM and [HCO_{3} ^{−}]_{i} = [HCO_{3} ^{−}]_{o} = 25 mM (patch solution b and bath solution D in Table 1), NBCe2C expressing HEK293 cells were voltageclamped at 50 mV and a series of voltage (including a pulse to 0 mV) was applied (Figure 4a, left panel). Increasing the Na^{+} concentration from 10 to 25 mM in the bath solution (bath solution E in Table 1) increased the voltagedependent current (Figure 4a, central panel). Net current (ΔI) through NBCe2C induced by changing [Na^{+}]_{o} was obtained by subtracting the currents in bath solution containing 10 mM Na^{+} from currents in 25 mM [Na^{+}]_{o} (Figure 4a, right panel). With this operation, according to Eq. 4, currents mediated by other channels and electrogenic transporters were eliminated if the two assumptions associated with Eq. 4 were satisfied. Figure 4b shows currentvoltage (IV) relation of steadystate current in bath solution containing 10 mM or 25 mM [Na^{+}]_{o} and Figure 4c shows ΔI of NBCe2C vs. voltages. Taking ΔI_{v1} at V = 0 and ΔI_{v2} at V = 12 mV, q is calculated using Eq. 7. We obtained q = 2.0 ± 0.14 (n = 5, Figure 4d). The results suggest that the transport stoichiometry ratio of NBCe2C is 2 HCO_{3} ^{−}: 1 Na^{+} (or 1 CO_{3} ^{2−}: 1 Na^{+}) in HEK293 cells. This result is consistent with the q value obtained with the conventional reversal potential method using the inhibitor DIDS (Figure 3).
Transport stoichiometry of NBCe1A estimated with the delta current method
Cells expressing NBCe1A were voltageclamped at 50 mV, and wholecell currents were recorded when a series of voltage pulses was applied (Figure 5a). Using the same conditions as above that [Na^{+}]_{i} = [Na^{+}]_{o} = 10 mM and [HCO_{3} ^{−}]_{i} = [HCO_{3} ^{−}]_{o} = 25 mM (patch solution b and bath solution D in Table 1), increasing the Na^{+} concentration from 10 to 25 mM in the bath solution (bath solution was switched from solution D to solution E of Table 1) increased voltagedependent current (Figure 5a middle panel). The net current (ΔI) through NBCe1A induced by changing [Na^{+}]_{o} (right panel of Figure 5a) was obtained by subtracting the current traces in the solution containing 10 mM [Na^{+}]_{o} from those in 25 mM [Na^{+}]_{o}. The currentvoltage (IV) relation of steadystate currents in bath solution containing 10 mM or 25 mM Na^{+} is shown in Figure 5b). Figure 5c shows ΔI of NBCe1A vs. membrane voltages. This was the result of operation of Eq. 4 and the currents mediated by other channels and electrogenic transporters were eliminated. Taking ΔI_{V1} at V = 0 and ΔI_{V2} at V = 12 mV, we calculated q using Eq. 7 for every cell. We determined q = 1.87 ± 0.062 (n = 6, Figure 5d). The results indicate that the transport stoichiometry ratio of NBCe1A is 2 HCO_{3} ^{−}: 1 Na^{+} or 1 CO_{3} ^{2−}: 1 Na^{+} in HEK293 cells. This estimate is consistent with our previous results using the conventional reversal potential method with DIDS [25].
Computational simulation: ΔI method estimates q accurately when there are additional conductances other than electrogenic NBC transport
In native tissue or expression systems such as oocytes or HEK293 cells, there are endogenous channels and electrogenic transporters other than the one under study. In these cases, the Δ current method is based on the assumption of additivity of membrane currents while the ΔE_{rev} method and its variations based on the assumption of additivity of reversal potentials [2],[15],[16]. Were the latter true, by altering the concentrations of the transported species, the contribution of other channels and electrogenic transporters could be subtracted and the relationship between delta E_{rev} and transported species concentrations and the transport stoichiometry easily obtained based on Eq. 1. This method, although widely used, is not consistent with GoldmanHodgkinKatz (GHK) theory [17],[18] where E_{rev} is a logarithmic function of sum of concentrations of ions inside and outside of the membrane; i.e. not additive.
Now, suppose there is one kind of channel that is permeable to a univalent ion with valence z_{s} and permeability of P_{s} on the cell membrane, in addition to an electrogenic NBC transporter. Based on Eq. 2 and the GHK current equation (with all original GHK assumptions applied [18]), the current would be
At V_{I=0} of the electrogenic NBC transporter plus one channel system
We can see that even with one additional channel, this equation contains more than one unknown such as K_{c}, P_{s} and ν_{HCO3}. What we measure in the electrophysiological experiments is V_{I=0}. V_{I=0} is a complicated nonadditive function of E_{NBC}. A simple expression for the relationship between stoichiometry and reversal potential is not obtained. We will see a similar situation when there is one additional electrogenic cotransporter transporting ions s1 and s2 with involved function K_{a}, valence Z_{s1} and Z_{s2}, stoichiometry ν_{s1} and ν_{s2} respectively:
Again, a simple expression for the relationship between stoichiometry and reversal potential is not obtained.
We performed a computational simulation of membrane currents and reversal potentials to show how a conductance in addition to electrogenic NBC transport affects the measurement of V_{I=0} and thus the estimate of q for this electrogenic NBC. Based on Eq. 2, currents were calculated with the same conditions as our wholecell patchclamp experiments for estimating q (delta current method above) of NBCe2C: [HCO_{3} ^{−}]_{i} = [HCO_{3} ^{−}]_{o} = 25, [Na^{+}]_{i} =10mM. Assuming q = 2, Figure 6a shows IV curves and V_{I=0}s when the bath solution switched from [Na^{+}]_{o} =10 mM to 25 mM and the delta current (ΔI). The stoichiometry ratios estimated either with the ΔE_{rev} or ΔI methods are equivalent when there was no conductance other than the electrogenic NBC transporter (Table 2). However, if a small Cl^{−} conductance (compared to the conductance of the NBCmediated current) was present, simulation with Eq. 8 showed that both V_{I=0} values at [Na^{+}]_{o} = 10 mM and [Na^{+}]_{o} = 25 mM shifted toward more negative value, but the shifts for the two conditions were different (Figure 6b). Therefore ΔE_{rev} differed from that obtained without the Cl^{−} conductance and leads to a different estimate of q = 2.17. When the Cl^{−} conductance was doubled, the estimate of q became 2.33 (Figure 6c). When we input q = 3 in the simulation, the estimate was 3 in the absence of any other conductance. After introducing either a small Cl^{−} conductance G_{Cl} or 2 x G_{Cl} (same as above), the estimate of q became 4.96 and 7.2 respectively with the ΔE_{rev} method (Figure 6d,e and f; note the insets; Table 2). However as shown in Table 2, the value of q determined using the ΔI method was unaffected by addition of a G_{Cl} on the membrane. Specifically, the ΔIV curves in the absence, presence of small or large G_{Cl} were identical. Therefore, the currents mediated by other channels had been eliminated in the procedure and had no effect on the estimation of q.
We then simulated NBCe1A transport in conditions similar to the proximal tubule cells in the rat kidney where the ionic concentrations (in mM) were [HCO_{3} ^{−}]_{o} = 24, [HCO_{3} ^{−}]_{i} = 13.4, [Na^{+}]_{o} = 150 and [Na^{+}]_{i} =17 mM [26]. In addition to NBCe1A, the Na^{+}/Dglucose cotransporter SGLT2 was modeled in the simulation. SGLT2 is expressed in the apical membrane of proximal tubule cells and exhibits a transport stoichiometry of 1 Na^{+}: 1 glucose [27]. One positive charge moves across the membrane per transport cycle. An extracellular glucose concentration [G]_{o} = 5 mM and intracellular [G]_{i} = 1 mM were substituted into Eq. 10 assuming q = 2 or 3 for NBCe1A. Table 3 shows the V_{I=0} values when [Na^{+}]_{o} = 150 and when [Na^{+}]_{o} was switched to 100 in the absence and presence of SGLT2. The simulation also provided estimated q values by ΔE_{rev} and ΔI methods. The stoichiometry ratios estimated either with the ΔE_{rev} or ΔI methods were equivalent when SGLT2 was absent. However, when SGLT2 was present, q was 2.55 estimated with the ΔE_{rev} method when the actual value in the simulation was 3 (Table 3). The presence of SGLT2 prevents any definitive determination as to whether the stoichiometry of NBCe1A is q = 2 or q = 3.
These results indicate that the ΔE_{rev} method can significantly bias the estimate depending on the magnitude and electrophysiological properties (e.g. the IV relationship) of other channels and electrogenic transporters if there are any, while the ΔI method gives a more accurate estimate of the transport stoichiometry q.
Discussion
In this study, we have demonstrated the development and utility of a new method for estimating the transport stoichiometry of electrogenic transport proteins. With this ΔI method, one subtracts the currents due to channels and transporters other than the one under study and thereby obtains the stoichiometry of the transporter without the need for a specific inhibitor. Using this method, we showed that the transport stoichiometry of the bicarbonate cotransporter NBCe2C expressed in HEK293 cells is 2 HCO_{3} ^{−}: 1 Na^{+} that is consistent with the results obtained using the conventional reversal potential method with the inhibitor DIDS. A transport stoichiometry ratio of 2 was also obtained for NBCe1A with the ΔI method that is consistent with the data obtained previously using the conventional reversal potential method with DIDS [25]. In addition, we demonstrated that, with computational simulation, the estimation of q obtained using the new ΔI method was equivalent to that obtained with the conventional ΔE_{rev} methods when an electrogenic NBC transporter was the only transport mechanism in the cell membrane. However, if a chloride channel or a glucose cotransporter SGLT2 was present in the membrane, our simulations showed that the ΔE_{rev} method significantly biased the estimate of the transport stoichiometry q, while the ΔI method gave accurate results.
The method proposed in this study is based on Eq. 2 from Heinz [23] that describes the functional relationship between flux of a transporter and the concentrations of transport ions/substrates and the membrane voltage [24]. Unlike the GHK formulation that assumes independence of ion movement across the membrane [13] and does not involve the concept of stoichiometry, Eq. 2 explicitly expresses coupling of Na^{+} and HCO_{3} ^{−} (both are voltage dependent) as a product and the stoichiometry as a power of the concentrations and voltage. Linearity of the current and voltage relation is not a presumption for Eq. 2 nor is it for the GHK equations [17],[18]. Nonlinearity of the IV curves results from: 1) the GHK equation is based on solubilitydiffusion theory. In GHK current equation, the current is an exponential function of the voltage. Similarly Eq. 2 shows that flux is an exponential function of voltage; 2) transport mechanisms of membrane channels or transporters represented by the permeability term Ps in GHK equations and K_{c} in Eq. 2 may be voltage dependent. With the conventional E_{rev} method, if the transporter under study is the only electrogenic pathway, this nonlinearity would not be a problem since the current is 0 and at this point, the voltage is the reversal potential under the conditions of the experimental substrate concentrations. However, if there are other channels or electrogenic transporters in the membrane and if a specific inhibitor is not available, V_{I=0} that can be measured is not the reversal potential for the transporter under study, but rather is the voltage at a point on the IV curve where the net result of the transporter current under study and currents mediated by other transporters and channels is 0. The alternative ΔE_{rev} method is problematic in that the assumption of reversal potential additivity is inconsistent with nonlinearity property of GHK equations and Eq. 2. This is solved by employing the ΔI method where the contribution of other channels or transporters can be eliminated without the assumption of E_{ver} additivity.
If we assume that an electrogenic NBC transporter has a fixed transport stoichiometry, if the only ions that cross the cell membrane are Na^{+} and HCO_{3} ^{−} , from Eq. 2 we have
When I = 0, we have
Therefore,
This is essentially Eq. 1 if we take ν_{Na} = 1 and q = ν_{HCO3}/ν_{Na}. Starting from here, the widely used delta reversal potential method to estimate stoichiometry [2],[15],[16] can be easily derived:
When we change Na^{+} concentration in the bath solution from [Na^{+}]_{o1} to [Na^{+}]_{o2}, we have
Then, delta reversal potential ΔE_{rev} would be
From the above operations, we can see that reversal potential method, the ΔE_{rev} method and the ΔI method to estimate transport stoichiometry all have the same theoretical foundation (such as Eq. 2 and same assumptions). Moreover, they are equivalent if the electrogenic transporter under investigation is the only conductive process in the membrane.
However, if there are endogenous channels and electrogenic transporters other than the one under study, the relationship of ion activities and transport stoichiometry and reversal potential becomes very complicated as we can see in Eq. 8, Eq. 9 and Eq. 10. Therefore, a method to eliminate the confounding effects of additional transporters and channels on reversal potentials by simple subtraction of V_{I=0} is not valid. Our simulation results also indicate that the commonly used ΔE_{rev} method in this instance would not be accurate. The error increases as the currents mediated by other transporters and channels increase (Table 2) relative to the transporter under investigation.
Transport parameters of an electrogenic secondary active transport like K_{c} are affected by many factors. How a given transport process responds theoretically to an electrochemical gradient depends on the type of the transport kinetic models utilized, e.g. “affinity model”, “velocity model” or “mixed model” as described by Heinz [24], and whether the loaded or the unloaded carrier bears an electrical charge. Heinz [23] originally introduced equation 2 and referred to K_{c} as a function of mobility and concentrations of the free and loaded carrier, respectively, and hence may vary with the degree of saturation. In our approach, we made two assumptions that are implicitly shared with the ΔE_{rev} method: 1) K_{c} is constant in certain voltage range and does not vary when the concentration of the substrate of choice in the study ([Na^{+}]_{o} in this study) changes; 2) the sum of currents $\sum _{j}{I}_{j}$ mediated by other channels and transporters in the membrane as a function of V does not change when the substrate concentration is altered [2],[15],[16]. Based on these two assumptions the two methods offer benefits such as experimentally straightforward as changing the concentrations of a substrate without the need for specific blockers and share similar limitations. The difference between ΔI and ΔE_{rev} method in terms of assumption 2 is that with the ΔI method, $\sum _{j}{I}_{j}$ can be completely eliminated (Eq. 4) if it does not change when the substrate ([Na^{+}]_{o} in this study) is altered. On the contrary, with the ΔE_{rev} method, as long as $\sum _{j}{I}_{j}$ is not negligible, the confounding effects of $\sum _{j}{I}_{j}$ on V_{I=0} can not be eliminated and biases the estimation of q as shown in Figure 6 and Table 2 and Table 3, even if it does not change when the substrate concentration varies.
In practice, ways to circumvent the limitations due to the above assumptions include: 1) using a smaller concentration change of the substrate, as long as it induces a significant delta current; 2) changing the concentrations of a particular substrate with less possibility of involving other electrogenic transporters. For example, in the case of electrogenic Na^{+}coupled glucose or amino acid transporters, one would choose to change either glucose or amino acids respectively rather than Na^{+}.
In this study, we changed [Na^{+}]_{o} from 10 to 25 mM because: 1) HCO_{3} ^{−} partakes in a volatile buffer system that involves pCO_{2} to keep the pH constant. pH would be stable when [HCO_{3} ^{−}]_{o} is unaltered; 2) switching [Na^{+}]_{o} from 10 to 25 mM would induce a significant delta current [15] and 3) at these relatively low concentrations, the possibility of transport saturation would be small, therefore variation of K_{c} in Eq. 2 and Eq. 3 would be minimized. We assigned V_{1} = 0 in the above application, therefore in the conditions of ${\left[N{a}^{+}\right]}_{i}={\left[N{a}^{+}\right]}_{o}\mathit{and}\phantom{\rule{0.25em}{0ex}}{\left[HC{{O}_{3}}^{}\right]}_{i}={\left[HC{{O}_{3}}^{}\right]}_{o},{I}_{M}={\displaystyle \sum _{j}{I}_{j}}$ is well defined and it is not close to 0. In addition, we assigned a V_{2} that is not far from 0 (+12 mV in this study), thus possible variation of K_{c} under extreme voltages can be minimized.
More detailed kinetic descriptions of the transport rate in order to characterize the entire IV relationship rely on a detailed understanding of the molecular transport steps [28][30]. This is not necessary for the purposes of our formulation, because we implicitly analyze the portion of the IV relationship that is close to the E_{rev} i.e., V_{1} = 0 when [Na^{+}]_{i} = [Na^{+}]_{o} and [HCO_{3} ^{−}]_{i} = [HCO_{3} ^{−}]_{o}.
The accuracy of stoichiometry estimation using wholecell patchclamp recordings also depends on the accuracy of wholecell current measurement and the voltages applied to the cell membrane from the patchclamp amplifier. The drift of the junction potential between the patch pipette solution and the Ag/AgCl coated wire that connects to the headstage of the amplifier is a major source of unstable current recording especially when the Cl^{−} concentration in the pipette is low [22]. We used a microagar salt bridge of 2 M KCl in the patch pipette that minimized the junction potential drift and therefore stabilized the wholecell current measurements [22].
Conclusions
We developed a new delta current (ΔI) method for estimating transport stoichiometry of electrogenic transporters based on a simplified model for electrogenic secondary active transport by Heinz (1981). We showed that this model reduces to the conventional reversal potential method when the transporter under study is the only electrogenic transport on the membrane. When there are other electrogenic transport processes such as ion channels or transporters, the ΔI method eliminates their contribution in estimation of q. We tested this new ΔI methodology in HEK293 cells expressing the electrogenic SLC4 sodium bicarbonate cotransporters NBCe2C and NBCe1A, as well as using computational simulations. Our simulations demonstrated that the ΔE_{rev} method introduces significant error when other channels or electrogenic transporters are present on the membrane with a significant conductance relative to the transporter under study, and that the ΔI equation accurately calculates the stoichiometric ratio. Our new ΔI method can be readily extended to the analysis of other electrogenic transporters.
Abbreviations
 CMV:

Cytomegalovirus
 DIDS:

4,4′Diisothiocyanatostilbene2,2′disulfonic acid
 EGFP:

Enhanced Green Fluorescent Protein
 Eq:

Equation
 E_{rev} :

Reversal potential
 F:

Faraday’s constant
 GHK:

GoldmanHodgkinKatz
 HEPES:

4(2Hydroxyethyl)1piperazineethanesulfonic acid
 IV:

Currentvoltage
 NBC:

Sodium bicarbonate cotransporter
 R:

Gas constant
 SGLT2:

Sodiumcoupled glucose transporter 2
 T:

Absolute temperature
 ΔE_{rev} :

Delta reversal potential
 ΔI:

Delta current
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Acknowledgement
We thank Dr. Donald D. F. Loo, for helpful discussions and comments on the manuscript.
This work was supported in part by funds from the NIH (R01DK077162), the Allan Smidt Charitable Fund, the Factor Family Foundation, and the Arvey Foundation (to IK) and R43DA03157801 (to X. M. S).
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XMS, IK conception and design of the research; XMS, and LK performed the experiments; XMS analyzed the data; XMS and IK drafted and revised the manuscript; XMS, LK, and IK approved final version of manuscript.
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Shao, X.M., Kao, L. & Kurtz, I. A novel delta current method for transport stoichiometry estimation. BMC Biophys 7, 14 (2014) doi:10.1186/s1362801400142
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Keywords
 Electrogenic transporter
 Stoichiometry
 Membrane currentvoltage relationship
 Reversal potential
 HEK293 cells
 Patch clamp
 Computational simulation