Simultaneous Determination of Low Free Mg2+ and pH in Human Sickle Cells using 31P NMR Spectroscopy* 210

The concentrations of free magnesium, [Mg2+]free, [H+], and [ATP] are important in the dehydration of red blood cells from patients with sickle cell anemia, but they are not easily measured. Consequently, we have developed a rapid, noninvasive NMR spectroscopic method using the phosphorus chemical shifts of ATP and 2,3-diphosphoglycerate (DPG) to determine [Mg2+]free and pH i simultaneously in fully oxygenated whole blood. The method employs theoretical equations expressing the observed chemical shift as a function of pH, K+, and [Mg2+]free, over a pH range of 5.75–8.5 and [Mg2+]free range 0–5 mm. The equations were adjusted to allow for the binding of hemoglobin to ATP and DPG, which required knowledge of the intracellular concentrations of ATP, DPG, K+, and hemoglobin. Normal oxygenated whole blood (n = 33) had a pH i of 7.20 ± 0.02, a [Mg2+]free of 0.41 ± 0.03 mm, and [DPG] of 7.69 ± 0.47 mm. Under the same conditions, whole sickle blood (n = 9) had normal [ATP] but significantly lower pH i (7.10 ± 0.03) and [Mg2+]free (0.32 ± 0.05 mm) than normal red cells, whereas [DPG] (10.8 ± 1.2 mm) was significantly higher. Because total magnesium was normal in sickle cells, the lower [Mg2+]freecould be attributed to increased [DPG] and therefore greater magnesium binding capacity of sickle cells.

Magnesium is the second most abundant intracellular cation after potassium and is important in the regulation of more than 300 enzymes (1)(2)(3) and ion transport across cell membranes (4,5). Some diseases, such as hypertension, pre-eclampsia, and sickle cell anemia exhibit pathologies linked to low magnesium levels (6 -8). The genetic defect in sickle cell anemia results in the synthesis of an abnormal ␤ hemoglobin (Hb) subunit, which polymerizes upon deoxygenation. This polymerization produces the characteristic sickle deformation in red blood cells and leads to microcirculatory occlusion and consequent tissue ischemia. Sickling depends strongly on the intracellular Hb concentration (9) and is enhanced by dehydration of erythrocytes, a process involving potassium and sodium transport across cell membranes, which is in turn thought to be regulated by cytosolic free magnesium, [Mg 2ϩ ] free , 1 via the KCl cotransporter (KCC1) (5,10). Thus dehydration of sickle cells, and thereby the process of sickling, may be controlled by changes in the magnesium content (8,11).
To assess its role in sickle cell anemia, an accurate and reproducible method was required for the measurement of [Mg 2ϩ ] free . Although several methods have been developed to determine [Mg 2ϩ ] free in red blood cells, such as null point titration with metallochromic dyes (12), magnesium-sensitive electrodes (13,14), use of divalent cation ionophores (15), and 31 P NMR spectroscopy (16,17), the reported values vary 3-fold, from 0.2 to 0.6 mM (18 -20). Of these methods, the one noninvasive technique that has been used extensively is that of 31 P NMR spectroscopy (16 -19, 21-25). However, there are a number of potential errors associated with using the observed NMR signal to determine [Mg 2ϩ ] free , including uncertainty in the binding constant between ATP and magnesium (26), the effect of intracellular ionic content on this interaction (27), and the effect of metabolite-Hb interactions (23). Gupta and co-workers (2), the first to use this method, addressed the first two problems by determining an apparent MgATP binding constant, K (app)bMgATP , from the difference between the end points of the chemical shifts of the magnesium bound and unbound ␣ and ␤ phosphorus nuclei of ATP, ␦ ␣␤-MgATP and ␦ ␣␤-ATP . This decreased the number of equilibria that had to be considered and overcame the need to rely on adjusted published values of K (app)b , which varied 100-fold (28). However, as with any simplification, problems are associated with this approach, which include the possible difference in ionic strength between solutions and in vivo conditions, the accuracy of the chemical shift end points required to determine K (app)b (29), and the assumption that ␦ ␣-ATP did not vary.
Because the chemical shifts of 31 P NMR phosphorus peaks depend on many different factors (30), accurate analysis should take all factors into account, which has become possible with the increased capacity of computation. Previous analyses using phosphorus chemical shifts have been used to obtain either pH (24) or [Mg 2ϩ ] free (16) in red blood cells. Only one approach used the 31 P resonances of ATP to determine both pH and [Mg 2ϩ ] free (31), but this averaged all ATP species with an apparent binding constant and made no allowance for the effect of Hb and K ϩ binding upon the observed chemical shift. It is possible that metabolite relaxation times become so long upon binding to Hb that such species do not contribute to the observed NMR peak, making them "NMR-invisible," and hence they could be omitted from analysis. Therefore, the effect of Hb on metabolite NMR "visibility" was investigated here. In addition, standard titration curves of ATP and DPG phosphorus chemical shifts were expressed in equations as functions of pH and [Mg 2ϩ ] free . Adjusting these equations for Hb and K ϩ binding to ATP and DPG improved the method for measuring pH and [Mg 2ϩ ] free in red blood cells. This newly developed NMR-based analysis allowed the simultaneous determination, for the first time, of the intracellular pH and [Mg 2ϩ ] free in normal and sickle blood. Preliminary reports of this work have been published in abstract form (32,33).

Preparation of Standard Solutions for Titration Curves-Solutions
were prepared as described previously (34) and contained 5 mM each K 2 ATP:2H 2 O, Na 5 DPG:3.5H 2 O (Sigma), and P i (British Drug House, Poole), at 310.15 K and an ionic strength of 0.25. The pH of the solutions ranged from 5.75 to 8.50, and the total magnesium varied from 0 to 15.9 mM. Ionic strength was adjusted to 0.25 by adding KCl, and the pH was adjusted using 1 M KOH or HCl. In total, 130 samples were prepared, with 13 pH values, at 10 [Mg 2ϩ ] free concentrations. Using the relevant equilibrium constants (Table I) where f is a ratio of the ionized ligand, A zϪ , to the total ligand, A T .
For the calibration solutions containing the three magnesium binding ligands, ATP, DPG, and P i , the [Mg] T required for each solution was calculated from Equation 3.
The solution of Equation 3 and the simultaneous equations involved in providing the correct ionic strength and pH (see Appendix I) were achieved using Mathematica™ (Wolfram Research, Champaign, IL). Using this method, MgCl 2 was added to give solutions containing [Mg 2ϩ ] free of 0, 0.05, 0.10, 0.25, 0.50, 0.75, 1.00, 1.25, 2.50, and 5.00 mM. 31 P NMR Spectroscopic Analysis of Standard Solutions-31 P NMR spectra of the solutions were acquired using a 9.4-tesla, Oxford Instruments wide bore superconducting magnet interfaced with a Varian Inova spectrometer operating at a phosphorus frequency of 161.9 MHz. A 10-mm probe was used, with the sample temperature set to 310.15 K. The homogeneity of the magnetic field was optimized by shimming on the 1 H free induction decay to give 1 H spectral line widths of 9 Ϯ 2 Hz. For each sample, a 90 o pulse (18.5 s) was used with an interpulse delay of 15 s, a spectral width of 8 kHz, and no proton decoupling. The delay time was based on preliminary experiments in which T1 values were determined for sample solutions by running a preinstalled macro on the Varian spectrometer, using a standard inversion recovery method. Each spectrum consisted of 128 summed transients. Up to 256 transients were acquired for samples in which the ␤-phosphorus of ATP was especially broad.
A capillary containing ϳ50 l of 0.5 M phenylphosphonic acid was used as an external chemical shift reference, standardized relative to phosphocreatine at 0.00 ppm. Prior to Fourier transformation, the signal:noise ratio was increased by multiplying the 31 P NMR free induction decays by an exponential function sufficient to generate a line broadening of 1 Hz, and the time domain signals were zero filled once. The NMR1 program (Tripos, St. Louis, MO) was used to fit peak areas, line widths, and chemical shifts of spectral resonances.
Theoretical Fitting of the Observed Chemical Shifts-The chemical shifts for the 3-phosphorus of DPG and ␤-phosphorus of ATP peaks were plotted against pH and [Mg 2ϩ ] free , generating a three-dimensional surface. This was then fitted to an equation using the nonlinear fit algorithm in Mathematica™ with the main parameters being the chemical shifts of the H ϩ , K ϩ , and Mg 2ϩ bound forms of ␤-ATP and 3P-DPG, and the binding constants of each species to H ϩ , K ϩ , and Mg 2ϩ .
The equation used for fitting was based on the principle that the observed chemical shift (␦ obs ) could be described by combining the chemical shifts of all of the relevant forms of a substance present in the equilibrium and weighting their contribution to the observed chemical shift according to the ratio of their population to that of the total substance present (35).
Hence for ATP, assuming all ATP to consist of several species, the theoretical chemical shift can be described by Equation 6.
Using the equilibrium constants (assuming activity coefficients of unity), the concentrations of all forms can be expressed relative to the concentration of the completely ionized form (f values). Taking these and substituting in the above equation, the observed chemical shift is represented by Equation 7.
Similarly for DPG, the following equation describing the observed chemical shift for DPG was formulated, assuming the forms present were DPG 5Ϫ , HDPG 4Ϫ , H 2 DPG 3Ϫ , MgDPG 3Ϫ , MgHDPG 2Ϫ , KDPG 4Ϫ , and KHDPG 3Ϫ . The substitution of [H ϩ ] to 10 ϪpH was used to make the equations more convenient. "Goodness" of fit may be determined by 2 with F i the value of the ith data point, and f i is the value obtained from the fit.
Model of the Red Blood Cell-The chemical shifts of the intermediate species were calculated from the above fitting equations and were used in new multiple equilibria equations, which included the binding of Hb to ATP 4Ϫ , MgATP 2Ϫ , and DPG 5Ϫ . To achieve this, it was assumed that the chemical shifts of each species were not affected by binding to Hb (2,14). Binding constants for Hb were taken from Berger et al. (36), where K bHbATP is 360 M Ϫ1 , K bHbMgATP is 39 M Ϫ1 , and K bHbDPG is 250 M Ϫ1 . Therefore, The above equations are simplified from the Mathematica program presented in Appendix II. After determination of chemical shifts of the ␤-phosphate peak of ATP (␦ ␤ ) and the phosphate on the 3 carbon of DPG (␦ 3P ) from acquired spectra of red blood cells, these equations were solved simultaneously to calculate pH and [Mg 2ϩ ] free in red blood cells using the Solve algorithm in Mathematica (Appendix II). A requirement of this method is that the total amounts of ATP, DPG, Hb, and K ϩ must be entered in the algorithm before a solution can be found. ATP and DPG concentrations were assayed as described below, and Hb and K ϩ concentrations (mmol/liter cell water) were assumed to be 7 mM (2, 37) and 160 mM (38), respectively. Erythrocyte total magnesium was calculated by summing the concentrations of all magnesium-containing species, determined from the solution of this algorithm, for comparison with the total magnesium determined using a standard colorimetric assay (see later).
Preparation of Whole Human Blood Samples-This study was approved by the Ethics Committee, Medical School, University of Birmingham. Blood from volunteers (8 -10 ml) or from consenting homozygous sickle cell disease (HbSS) patients (5 ml) was collected by venipuncture into lithium-heparinized syringes. To simulate in vivo fully oxygenated blood, whole blood samples were prepared for analysis by equilibration with a mixture of O 2 and CO 2 (95%:5%), adjusted to give a final P CO2 of 40 mmHg and P O2 between 200 and 400 mmHg, determined using a blood gas analyzer (Radiometer ABL 330). The PO 2 , PCO 2 , and pH ex were measured before and after NMR analysis, at which time the PO 2 remained Ͼ 150 mm Hg, to ensure complete oxygenation at all times during analysis. To minimize cell degradation, time from phlebotomy to analysis was kept to a minimum (1-3 h), during which blood samples were stored on ice.
Separation of Sickle Cells by Density-Sickle cells are heterogeneous with respect to total magnesium, DPG content, and pH (39,40), which vary with cell density as the cell sickles, therefore we separated sickle cells into three fractions using a discontinuous arabinogalactose density gradient before analysis, thereby allowing comparison with unfractionated whole sickle blood. Cells were fractionated on discontinuous density gradients of Stractan II (arabinogalactan) as described previously (41) with minor modifications (42). 2 All separations were performed at 4°C in a cold room to maintain ATP concentrations. Briefly, two isoosmotic Stractan density layers, 1.090 and 1.099 g/ml, were used with a cushion of 1.196 g/ml. Blood (20 ml) was collected from six consenting HbSS patients, unmatched for blood antigen type (one with blood group A, one with blood group O, and four with blood group B). Following centrifugation at 3,000 ϫ g for 10 min, plasma was removed and stored on ice, and the buffy coat was carefully removed and discarded. Each of the six packed blood cell samples was washed twice with ice-cold buffer containing 5 mM KCl, 145 mM NaCl, 0.05 mM EGTA, 0.2 mM MgCl 2 , 10 mM Na-HEPES, pH 7.4. For each experiment, cells from two patients were pooled to give a total of 8 -10 ml of packed cells, which were layered on the Stractan gradient. After ultracentrifugation, three layers were harvested, light (d Ͻ 1.090 g/ml), medium (1.090 g/ml Ͻ d Ͻ 1.099 g/ml), and dense (d Ͼ 1.099 g/ml), with the dense layer containing most of the irreversibly sickled cells. The density-fractionated red cells were washed three times at 4°C, using the above buffer, before being resuspended in pooled, heparinized plasma from the same two patients, to give final hematocrits of 35-50%. Normal ATP concentrations (see later) and microscopic observation at high magnification showed that no agglutination occurred in the pooled, unmatched blood samples, possibly because of the high heparin concentrations (43). The "whole blood" samples were prepared for NMR analyses as described above. The time taken from phlebotomy to NMR analyses was 5-7 h. 31 P NMR Spectroscopy of Human Red Blood Cells-31 P NMR spectra of the cells were acquired using the magnet and spectrometer described above. The fully oxygenated whole blood (ϳ 4.5 ml), prepared as above, was sealed in a 10-mm NMR sample tube together with an external reference capillary containing 50 l of 0.3 M phenylphosphonic acid and brought to 37°C before 31 P NMR spectroscopic analysis in a 10-mm probe. The probe temperature was set to 310.15, the samples were not spun, and no proton decoupling was used. The homogeneity of the magnetic field was optimized by shimming on the 1 H free induction decay to a line width of 30 Ϯ 2 Hz (normal cells) or 23 Ϯ 2 Hz (sickle cells). A 45°pulse was used with an interpulse delay of 0.52 s to give peaks of a high signal:noise ratio (25). The spectral width was 8 kHz, with an acquisition time of 0.272 s (4,500 data points), and 2048 transients were summed. The samples were at 37°C for a maximum of 38 min. Prior to Fourier transformation, the signal:noise ratio was increased by multiplying the 31 P NMR free induction decays by an exponential function sufficient to generate a line broadening of 7 Hz, and the time domain signals were zero filled once, giving a digital resolution of 2 V. L. Lew, personal communication.
1.1 Hz/point. The NMR1 program was used to fit peak areas, line widths, and chemical shifts of spectral resonances. Chemical Assays-Immediately after the NMR measurement, the blood was poured from the NMR tube and samples taken for light microscopy to count the number of sickled cells, and for determination of hematocrit and ATP and DPG concentrations using diagnostic kits (Sigma). Total magnesium concentrations in plasma and trichloroacetic acid extracts of whole blood were determined in duplicate using diagnostic kits (Sigma). The total erythrocyte magnesium concentrations were then calculated by subtracting the plasma total magnesium from the whole blood measurement. Concentrations are presented per liter of cell water, assuming a 95% packing efficiency and 70% cellular water content for normal cells, 66% for whole sickle blood, and 65 and 62% for medium and dense sickle cells, respectively (37). The hematocrits were determined after centrifugation at 13,000 ϫ g for 10 min in a microhematocrit centrifuge using a Hawksley microhematocrit reader, with all measurements made in triplicate.
Effect of Hb on ATP and DPG NMR Visibility-To alleviate concerns that metabolites bound to Hb would be NMR-invisible, solutions were prepared containing 2 mM K 2 ATP:2H 2 O, 7 mM Na 5 DPG:3.5H 2 O (Sigma), 190 mM KCl, 3 mM MgCl 2 :6H 2 O with and without 6 mM human Hb (Sigma), and the pH was adjusted to 7.2 at 37°C using 1 M KOH or HCl. 31 P NMR spectra of the solutions were acquired as described above. Immediately after the NMR measurement, ATP and DPG concentrations were determined as described above.
Statistics-Data are presented as means Ϯ S.D. Significance was tested using analysis of variance, with repeated measures where appropriate, with p Ͻ 0.05 considered significant.

RESULTS
Typical 31 P NMR spectra of the titration solutions acquired between a pH 6 of 8 and [Mg 2ϩ ] free between 0 and 5 mM are shown in Fig. 1. The ATP, DPG, and P i peaks increased in chemical shift as pH increased, reflecting a decrease in electronic shielding upon proton association. The maximal chemical shift changes are given in Table II. Increasing [Mg 2ϩ ] free altered the ␦ of all ATP peaks (␤ (3.26 ppm) and ␥ (3.72 ppm) more than ␣ (0.46 ppm)) but led to only slight changes in those of P i (0.03 ppm) and 3P-and 2P-DPG (0.34 and 0.76 ppm, respectively). The chemical shifts of the 2P-DPG and P i peaks were more sensitive to changes in pH than to changes in magnesium (3.15 and 2.35 ppm, respectively), whereas those of the ␤and ␥-phosphate peaks of ATP were equally sensitive to both pH and [Mg 2ϩ ] free . At the physiological pH of 7.2 and [Mg 2ϩ ] free of 0.3 mM, the ␤-ATP triplet could not be fully resolved because of line broadening caused by chemical exchange, but the peak could still be used for chemical shift determination. The P i peak overlapped with the 2P-DPG doublet, thus neither could be used for pH or [Mg 2ϩ ] free determination in red blood cells. Instead, it was necessary to use the 3P-DPG peak.
Three-dimensional Curves from the Chemical Shifts of ␤-Phosphate of ATP and 3-Phosphate of DPG-Using the ob- served ␤-ATP peak chemical shifts (␦ ␤ ), a three-dimensional surface curve was generated as a plot of pH and [Mg 2ϩ ] free with the section relevant to physiological conditions expanded (Fig. 2a). Equation 7 was fitted to these data and plotted in Fig.  2b. Using the observed 3P-DPG peak chemical shifts (␦ 3P ), a three-dimensional surface curve was generated as a plot of pH and [Mg 2ϩ ] free (Fig. 3a). Equation 8 was fitted to these data and plotted in Fig. 3b. The final values of the fitting parameters of chemical shifts and explicit equilibrium constants are given in Table III.
Effect of Hb on ATP and DPG NMR Visibility-The line widths of DPG and ATP peaks each increased by 60% on addition of Hb (␣-LW from 28 to 45 Hz, and ␤-LW from 50 to 80 Hz). Despite this, when the normalized peak integral was divided by the assayed total metabolite concentration, the values changed by less than 3% on addition of Hb. This indicated that ATP and DPG NMR visibility was not significantly impaired by binding to Hb, in that the Hb-bound component of the observed chemical shift was not so broad as to be lost in the noise of the spectrum. Fig. 4 shows a typical 31 P NMR spectrum of normal red blood cells. The DPG peaks were relatively sharp (LW Х 20 Hz), whereas the line width of the ␤ phosphate of the ATP peak was significantly greater than the line widths of the other ATP peaks (␤-LW Х 65 Hz, ␣-LW Х 35 Hz). Using the 3P-DPG and ␤-ATP phosphate peak chemical shifts, the intracellular pH and [Mg 2ϩ ] free were determined simultaneously using the values of ATP and DPG determined by enzymatic analysis.
Analysis of Normal and Sickle Red Blood Cells-Fractionation of the 8 -10 ml of packed sickle cells gave fraction volumes in the approximate ratio 3:5:2 for light:medium:dense fractions. When resuspended in their plasma, the hematocrits were in the range of 35-50%. Morphological microscopic examination, as estimated by criteria described previously (42), revealed that the dense fraction contained between 50 and 70% irreversibly sickled cells, the medium contained ϳ5%, and the light had fewer than 1%.
The determined levels of DPG, ATP, pH i , [Mg 2ϩ ] free , and Mg T , both predicted using the model and assayed, are presented in Table IV for normal and sickle whole blood and for each of the separated sickle cell fractions, light, medium, and dense. DPG was elevated significantly in all three sickle cell fractions, by a maximum of 2.4 mM in the medium density fraction, compared with normal blood cell DPG of 7.7 mM. However, the DPG concentration of the dense cells was significantly lower than that of the other sickle cells, but not as low as the normal cells. ATP was normal in all of the sickle cell samples. The PCO 2 of whole blood and cell fractions was set to be the same for all the analyses, which gave normal blood a measured pH ex of 7.39 Ϯ 0.03. However, at a PCO 2 of 40 mm Hg, the pH ex of whole sickle blood was 0.07 pH unit more acidic than normal blood. After fractionation, the pH ex of sickle cells was 0.23 pH unit more acidic than whole control blood. The PO2 never dropped below 150 mm Hg and usually was around 300 mm Hg. In parallel with their more acidic pH ex , the pH i of all sickle cell samples was 0.10 -0.13 pH unit more acidic than normal cells. [Mg 2ϩ ] free was decreased significantly by ϳ0.09 mM in the medium and the dense cells, and in the whole sickle blood compared with normal cells. There was no change in Mg T , either predicted from the model or measured by assay.
Thus, using this new method, which considered all significant interactions between metabolites, H ϩ , K ϩ , Mg 2ϩ , and Hb, oxygenated sickle blood had normal ATP concentrations and significantly lower pH ex , pH i , and [Mg 2ϩ ] free than normal cells, both in the dense fractions after separation and as unfractionated whole blood. DISCUSSION Less than 0.5% of body magnesium resides in blood plasma, and yet plasma total magnesium is commonly used as a marker of body magnesium status because of the absence of an easy reproducible method for measuring [Mg 2ϩ ] free. Because intracellular [Mg 2ϩ ] free is important in regulating biochemical reactions, it may serve as a better marker of status, and, of all tissues, red blood cells are the most readily available for study in humans. The best method by which to measure [Mg 2ϩ ] free is frequently discussed, but the effect of pH on [Mg 2ϩ ] free measurements is often not taken fully into account (27 The equations relating the observed chemical shift of 3P-DPG and ␤-ATP and to pH and Mg 2ϩ accurately fitted both experimental curves using the theoretical equations (Equations 7 and 8). These equations were solved simultaneously to determine both intracellular pH and [Mg 2ϩ ] free in normal and sickle red blood cells (Table IV). 31 P NMR studies (44) on the effects of Mg 2ϩ on ATP have shown that Mg 2ϩ induces the greatest chemical shift on the ␤-phosphorus and the least on the ␣-phosphorus resonance of ATP. This has been interpreted as indicating that Mg 2ϩ binds mainly to the ␤and ␥-phosphate groups (45,46), possibly via a cyclic six-membered transition state that is energetically favored (Fig. 5) and has little interaction with the ␣ residue. Thus, previous studies have used the changing difference of the chemical shift of the ␤ from the ␣ resonance of ATP, ␦ ␣␤ , to determine intracellular [Mg 2ϩ ] free in many tissue types (16,47,48) and thereby use the ␣-ATP peak as an internal chemical shift standard. However, it was seen (Fig. 1) that the ␣-phosphate chemical shift was not only variable over the ranges of pH and magnesium studied, but also ␣-phosphorus and ␤-phosphorus changed as pH and magnesium were varied. Therefore, in this study, all chemical shifts were measured relative to an external capillary standard.
Use of Explicit Equilibrium Constants-One important new aspect of this study was the use of explicit equilibrium constants as opposed to apparent constants, which allows the ready adaptation of the technique to a much wider range of conditions, either in red blood cells or other tissues. However, it    (29). Third, rather than absolute chemical shifts, the difference between ␦ ␣-ATP and ␦ ␤-ATP was used to determine K (app)dMgATP , assuming that neither was particularly affected by pH near neutrality. However, this is not the case at extreme magnesium levels where binding constants are determined. One further limitation of this approach, although it might, in theory, alleviate the need to know the exact effect that intracellular ionic content would have on binding interactions, is that it places restrictions on how readily the method can be applied under a range of conditions. For each new intracellular condition, including a pH change, a new apparent binding constant would have to be determined experimentally.
Effect of K ϩ -Different physiological conditions may have different potassium concentrations. Changing [K ϩ ] exerts not only an effect on K bMgATP , because of changing the ionic strength, but also an effect on the observed chemical shift by changing the amount of potassium-bound species (27). By assuming KX and KHX species existed for both DPG and ATP, a significant improvement in the mathematical fit of the observed data occurred, with 2 decreasing by 50 and 90% for DPG and ATP curves, respectively (data not shown). As a verification of this model, the new value for the explicit MgATP 2Ϫ binding constant of 7.50 ϫ 10 4 M Ϫ1 was in excellent agreement to that calculated by Mulquiney and Kuchel (37), which approximated to 8.0 ϫ 10 4 M Ϫ1 when adjusted for pH and ionic strength.
Effect of Hb on Metabolite NMR Visibility-If the metabolite relaxation times were sufficiently long when bound to Hb, the Hb-bound component of the overall observed chemical shift would be broad enough to be lost in spectral noise. Thus, Hb-bound metabolites would be NMR-invisible and could be omitted from the analysis. However, we found no loss in metabolite visibility on binding to Hb, so the effect of oxygenated Hb binding must be included in the analysis of [Mg 2ϩ ] free . Furthermore, when the concentration of ATP in red blood cells was estimated from the relative peak integrals of ATP and DPG in the observed spectra using the assayed concentration of DPG, there was less than a 10% variation between this value and that of the spectrophotometrically measured ATP concen-tration. This confirms the finding that DPG and ATP remain NMR-visible when bound to Hb.
Effect of Hemoglobin Binding on [Mg 2ϩ ] free Analysis-In the original 31 P NMR method, the effect that metabolite-Hb interactions had on the equilibria was considered (16). However, by 1983 (48), this complexity in the method was lost, perhaps because, according to their equilibrium constants, Hb bound MgATP and ATP equally strongly, which would allow the Hb effect to be ignored (23). Also, this added complexity, combined with the lack of computational power at the time, perhaps made solving the necessary multiple equilibria too difficult to be achieved routinely. A disregard of the effect of Hb has existed ever since, until 1997 when the effect of metabolite-Hb interactions was reevaluated (23).
The standard titration solutions did not include Hb because the extent of DPG and ATP binding to Hb was uncertain, and its effect would complicate the analysis of the important interactions among H ϩ , Mg 2ϩ , K ϩ , ATP 4Ϫ , and DPG 5Ϫ . However, after the observed experimental chemical shifts were modeled by equations describing the effect of all intermediates as functions of H ϩ and Mg 2ϩ , these equations were altered to allow for Hb binding (Equations 10 and 11). Although each equilibrium is independent of every other, and therefore the addition of Hb cannot, per se, alter the magnesium binding ATP equilibrium, the observed chemical shift is not independent because it intrinsically reflects a weighted average of all species present. Therefore the effect of Hb binding on the observed chemical shifts of both DPG and ATP must be taken into account. This can be achieved, to a first approximation, by assuming that chemical shifts did not change when metabolite species bound to Hb. This is true within a 5% error in oxygenated erythrocytes (2,23). Simultaneous equations were then formulated which included the effect that Hb binding had on the equilibria for ATP and DPG and hence on the overall observed chemical shifts (Equations 10 and 11 and Appendix II). These equations also required the binding constants for Hb binding to DPG, ATP, and MgATP. There are two generally accepted sets of constants (2), which largely agree except on the value of MgATP binding to Hb. This affected the positions of the equilibria to a different extent as illustrated in Fig. 6, a and b) for the analysis of pH and [Mg 2ϩ ] free with a total ATP of 2 mM, a total DPG of 5 mM, and a total Hb of 7 mM.
This difference led to discrepancies in the calculated values of pH i and [Mg 2ϩ ] free . For comparison, for normal blood, pH, and [Mg 2ϩ ] free were determined to be 7.23 Ϯ 0.01 and 0.41 Ϯ 0.04 mM, respectively, using Berger's constants (36), but 7.25 Ϯ 0.01 and 0.21 Ϯ 0.03 mM using those of Gupta et al. (2). The analysis presented under "Results" used Berger's constants because errors may be associated with the methodology (23) used by Gupta (16).
Effect of K ϩ and Hb-For a complete description of the erythrocyte, the effects of both K ϩ and Hb should be taken into account. However, once K ϩ is specified in the magnesium-ATP equilibrium, it also needs to be specified in the ATP-Hb equilibrium. Both Berger (36) and Gupta (2) measured apparent Hb binding constants using a physiological [K ϩ ], and therefore there are no data available on binding between KATP/KHATP and Hb. However, if realistic binding constants were estimated such that the overall binding of "ATP" (ATP free , HATP, KATP, KHATP) to Hb remained equal to that measured by Berger, a realistic possible adjustment to the theoretical equations can be made. Including all effects of both K ϩ and Hb, this provided a complete model of erythrocyte conditions and led to an estimation of pH and [Mg 2ϩ ] free in normal red blood cells of 7.20 and 0.41 mM, respectively (Table IV) partly verified by summing all MgX species to find the model prediction for erythrocyte total magnesium. This was calculated to be 3.1 mM, which is ϳ90% of that found experimentally. However, the model does not include the low affinity binding of Mg 2ϩ to Hb, which, it has been suggested, can account for the remaining 10% (49). Because Mg 2ϩ does not compete with ATP and DPG in binding Hb, this factor did not significantly change the analysis for [Mg 2ϩ ] free .
Comparisons with Previous Methods-In the most common analysis of NMR spectra to yield [Mg 2ϩ ] free (16,48,50), ␦ ATP and ␦ MgATP end points and an apparent dissociation constant are determined from in vitro solutions. It is then assumed that pH i is 7.2.
Following this procedure, using end points determined here and assuming K (app)dMgATP ϭ 38 M (51), [Mg 2ϩ ] free was calculated to be 0.24 Ϯ 0.02 and 0.16 Ϯ 0.02 mM for normal and sickle cells, respectively, significantly different from the values of 0.41 Ϯ 0.03 and 0.32 Ϯ 0.05 mM found in this work (Table V). Also, a comparison was made with values calculated using the original technique (16), where pH i is, again, assumed to be 7.2, but Hb interactions are included. This also leads to significantly different values.
Use of Explicit Constants for the Generation of in Vitro Solutions-Solutions are often required in experimental practice to mimic the conditions found in vivo. These frequently require an accurate [Mg 2ϩ ] free in ATP-containing solutions, but all too often this is calculated using equilibrium constants that are inappropriate for the conditions used. This is partly because of constants being quoted as apparent ones, thus making it difficult to adjust them over a range of conditions. However, in this work, explicit constants have been determined which readily allow the generation of a specific [Mg 2ϩ ] free in an ATP-containing solution over any range of pH, potassium, ATP, DPG, Hb, temperature, and ionic strength. Furthermore, this technique can be used in other cell types if a replacement peak chemical shift is used instead of DPG, ideally P i .
Normal   Ortiz et al. (40) reported an increased [Mg 2ϩ ] free and decreased Mg T . This elevation in [Mg 2ϩ ] free was markedly exacerbated on deoxygenation, reversing the usual inward magnesium concentration gradient and providing a mechanism for the magnesium depletion that they observed in this fraction. The difference in oxygenated dense cells between the two studies could be attributed to the difference in the densities chosen for cell fractionation and/or a difference in magnesium buffering. Although the blood samples were gassed with 95% O 2 to produce a PO 2 Ͼ150 mm Hg, it is possible that the dense sickle cells were not fully oxygenated, causing sickle Hb (Hb S) to polymerize (52). To our knowledge, there are no constants available for binding of metabolites to Hb polymer, but polymerization would alter [Mg 2ϩ ] free estimated using our method and may explain the low [Mg 2ϩ ] free in the dense cell fraction.
We found that the pH i of whole sickle blood was 0.10 pH unit more acidic than that of normal red cells, with a pH i of 7.10, in agreement with previous NMR studies of buffer-washed red cells (24). However, at the same PCO 2 , sickle blood pH ex was also significantly more acidic, by 0.07 pH unit, than normal.
Low [Mg 2ϩ ] free and low pH will both activate KCC1 (53), which may partially explain the high activity of this transporter in sickle cells. Cell shrinkage, perhaps via overactivity of KCC1, remains the principal hazard in irreversible sickling. For a patient with sickle cell anemia in a steady state, the percentage of red blood cells that are irreversibly sickled is about 15%. With a hematocrit of 20%, this corresponds to 3% of total blood volume. Therefore, although it is important to fractionate cells and analyze the dense cell fraction to understand the mechanisms underlying cell sickling, it is clinically important to analyze whole sickle blood, using considerably smaller volumes, to assess the in vivo state of the majority of red blood cells in an effort to prevent sickling.
In summary, we have modeled the observed chemical shift of ␤-ATP and 3P-DPG over a wide range of pH and [Mg 2ϩ ] free using appropriate theoretical equations. These equations included contributions to the overall chemical shift from all possible intermediates present in solution and represented the multiple equilibria formed by the interactions among H ϩ , Mg 2ϩ , K ϩ , ATP 4Ϫ , and DPG 5Ϫ . An adjustment for Hb binding was made to the equations to allow a new, more complete, determination of intracellular pH and [Mg 2ϩ ] free using 31   a Method 1 is the most commonly used method, in which ␦ ATP and ␦ MgATP end points are calculated from in vitro solutions, along with an apparent MgATP dissociation constant, K (app)dMgATP . It is then assumed that pH ϭ 7.2 and [Mg 2ϩ ] free ϭ K (app)dMgATP ϫ (␦ ATP Ϫ ␦ obs )/(␦ obs Ϫ ␦ MgATP ).
b Method 2 is the technique used by Gupta et al. (16), which includes interactions of DPG and ATP with hemoglobin using the authors' association constants, and assumes pH to be 7.2.
c Method 3 is an analysis using equations that include all possible interactions between H ϩ , K ϩ , Mg 2ϩ , Hb, ATP 4Ϫ , and DPG 5Ϫ , using parameters from Table III