INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX

A SIMPLE MODEL of acid-base equilibria in biologic fluids was described in 1981 by Stewart (14); this model has been shown to predict the concentrations of, e.g., H⁺ ([H⁺]) and HCO₃ ([HCO₃]) over a wide range of conditions (9, 18). Stewart explicitly included a single weak acid term to represent the role of all the weak acids found in plasma (mainly protein and phosphate). The role of plasma proteins in acid-base balance has been appreciated since the early studies of Van Slyke et al. (17), and this work was extended by the more recent studies of Fencl and collaborators (4-6, 12, 13). In most proteins, several of the amino acids contribute to determining the acid-base status, so several dissociation constants are involved. Stewart's approach was to use a single dissociation constant (K_A) and a net maximum negative charge concentration (A_Tot). In addition to this simplification, the values of K_A and A_Tot chosen by Stewart (14, 15) were not well documented. In 1981, K_A = 2.0 × 10⁷ and A_Tot = 0.020 eq/l were used. In Stewart's 1983 review, K_A = 3.0 × 10⁷ eq/l and A_Tot = 0.019 eq/l were used with no obvious justification. Others continue to use the latter values (9).

Rossing et al. (13) showed how changing plasma protein concentration in vitro could create an alkalosis or an acidosis similar to those seen in clinical protein concentration disturbances. This was followed by further work in which Figge, Rossing, and Fencl (6) looked closely at the treatment of the plasma proteins as weak acids. Using tonometry data from human serum albumin solutions, they established the pH response of an albumin-electrolyte mixture to a wide range of PCO₂ and strong ion differences (SIDs). They then used the known amino acid composition and the dissociation constants of these acids to build a very detailed acid-base model of human serum albumin. The model included the 16 histidine residues that are critical to the acid-base behavior in the physiological range (6.8-7.8) (10). In the following year, Figge, Mydosh, and Fencl (5) modified the model by using better estimates of the imidazole dissociation constants. By comparing the new model predictions with the previously reported tonometry data, they were able to determine estimates of four model parameters that could not be obtained directly: three missing pK values and the number of lysine and arginine residues having a pK of 11. This completed model (here labeled the Fencl model) gave an excellent fit to the albumin data. The Fencl model was also used to predict the pH of normal human plasma that had been similarly subjected to a wide range of PCO₂ and SID conditions, and, again, the pH predictions were very close to the measured values (5).

On the basis of the findings of Van Slyke et al. in 1928 (17), the net charge on the plasma proteins is often calculated as 16-17 meq/l. However, this is probably too high for the reason that the horse globulin samples used by Van Slyke et al. were contaminated by albumin (6, 16). The Fencl model predicts a net charge of 11.4 meq/l, which is very close to the value found in the horse albumin data of Van Slyke et al. and to the 12.2 meq/l found by Van Leeuwen (16) in plasma protein samples from normal humans.

Figge, Rossing, and Fencl (6) also clarified how to account for changes in plasma protein concentration. By showing that the nonalbumin proteins could be neglected, they also showed that the total protein charge could be expressed on a molar basis. In terms of a single-dissociation-constant model of albumin, A_Tot for plasma may be calculated from

(1)

where Z_Alb is the charge per mole of albumin, [Alb] is the concentration of albumin (in g/dl), and 66,500 is the molecular weight of albumin. From Stewart's values for A_Tot and with the assumption that [Alb] is 4.2 g/dl, the value of Z_Alb implied by Stewart is ~30 eq/mol.

The Fencl model also separated the three dissociations of phosphoric acid from that of protein to create a detailed and full treatment of plasma. Consequently, the Fencl model can accommodate such disease conditions as renal failure, where hyperphosphatemia and hypoproteinemia may exist simultaneously.

The Stewart and Fencl models were compared in 1994 by Kowalchuk and Scheuermann (9) using their own exercise data from human subjects. They showed that both models were able to predict [H⁺] and [HCO₃] equally well, although the comparison was weakened by the scatter in the predictions of both models, especially at high values of [H⁺]. The finding of Kowalchuk and Scheuermann suggests that the single K_A approach of the Stewart model may be adequate for some purposes. However, the approach still suffers from an arbitrary choice of K_A.

Further insight into the role of protein in acid-base chemistry has come from the temperature studies of Reeves (10). He and others argued that not only does histidine have a pK close to 7, it also has a temperature coefficient very similar to that of the ion product for water (K'_w), an observation that has important implications for the numerous functions of proteins and that has led to the alphastat hypothesis (10). The importance of histidine suggests that if a single-dissociation-constant model is useful, perhaps the K_A of choice should be the constant of histidine K_H.

There is another weakness in the Stewart model. Stewart chose to represent the overall behavior of the plasma weak acids by

(2)

where [A] and [HA] are the concentrations of weak acid anion and weak acid, respectively. From this and the mass balance for the weak acids, the amount of weak acid anion was given by

(3)

However, serum albumin, by far the most important of the weak acids in plasma acid-base balance (6), does not have this type of chemistry in the physiological range (pH 6.8-7.8). The protein contains ~99 amino acids (5) that maintain an almost fixed negative charge (mainly aspartic and glutamic acids) and ~77 residues (5) (mainly arginine and lysine) that keep an almost fixed positive charge in this pH range. Hence, there is net fixed negative charge (A_Fixed) of ~22 eq/mol. Albumin also contains 16 histidine residues (5), and the imidazole group on histidine reacts with H⁺ in the following manner

(4)

From mass balance, it follows that

(5)

where [A_H,Tot] is the concentration of histidine residues. Therefore, a simple but justifiable model for the concentration¹ of the net charge on serum albumin in the physiological range is

(6)

Stewart's model of weak acids (Eq. 3) has two parameters, A_Tot and K_A, the values of which cannot be determined from the physicochemical properties of protein and phosphate. Equation 6 has three constants to be determined, but all can be determined directly. As mentioned, A_Fixed should be close to 21 eq/mol, and A_H,Tot should be ~16 mol/mol. Determining K_H, however, is less straightforward, because the individual histidine moieties are in different microenvironments that influence the dissociation (1, 10). It has been found that the 16 histidines in albumin have 16 pK values with a range of 4.85-7.49 (1). However, Reeves (10) measured the pH dependence of plasma protein solutions at various temperatures and calculated the average dissociation of the histidine components. From his data, K_H is 1.77 × 10⁷ eq/l at 37.5°C, completing the determination of the parameter values in Eq. 6. It is therefore possible to build a model of acid-base chemistry for plasma and other solutions that follows the Stewart principles without the weaknesses of the original Stewart analysis. The purpose of this communication is to show that such a model, with the equations describing protein dissociation and with the parameters entirely determined by the measured chemical properties of serum albumin, can accurately predict the pH of albumin solutions and dialyzed human serum.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX

Two mathematical models of acid-base balance in isolated plasma were constructed and explored. The first model was identical to that described by Figge, Mydosh, and Fencl (5). The second model (labeled the single-association-constant model) was identical to the Fencl model, except [A] was calculated from Eq. 6 instead of from the multiple dissociations listed in Tables II and III of Ref. 5. Both models included the three dissociations of phosphoric acid separately from the protein chemistry. The pH predictions of both models were compared with the measured pH values in the detailed tonometry data sets published by Figge, Rossing, and Fencl (6).

Model Equations

(7)

(8)

The net number of "fixed" charges per mole of albumin was initially determined by summing the positive and negative charges over a range of pH by using the amino acid composition and dissociation constants given by Figge et al. (6) (Table 1). Histidine is the only albumin amino acid participating in acid-base equilibrium not included in Table 1. That analysis showed that the net number of fixed charges per mole of albumin was between 21 and 23 eq/mol. However, it is not possible to determine exactly the number of amino acids involved in pH regulation, because the tertiary structure of the protein may shield some sites from the surrounding fluid. Figge et al. (5) found the appropriate number of arginine and lysine residues by determining the number of these components that best fit the data. Similarly, to select the most appropriate number of fixed charges, the effect of changing the number from 15 to 26 eq/mol was investigated. A fixed charge of 21 eq/mol best fit the albumin data (see below).

With use of this value, the concentration of A_Fixed ([A_Fixed]) was calculated from

(9)

where [Alb] is the albumin concentration (in g/dl), 66,500 is the molecular weight of serum albumin, and 21 is the number of net negative charges per mole.

The concentration of A_H,Tot ([A_H,Tot]) was determined from

(10)

where 16 is the number of histidine residues per mole. K_H was determined from the polynomial given by Reeves to be 1.77 × 10³ eq/l at 37.5°C. The sensitivity of the predictions to the value of K_H was assessed by calculating the predicted pH values when K_H was 1.0 × 10³ and 3.0 × 10³ eq/l.

Other Details

For both models, the equations were solved simultaneously using the algorithm described by Stewart (14). The programs were written in Turbo Pascal and executed on a desktop computer. The pH predicted by a particular model was calculated for each of the 65 filtrands of Table A and 72 filtrands of Table B. For example, by using the data of filtrand 1, the Na⁺, K⁺, Ca²⁺, Mg²⁺, and Cl data were combined to give an SID of 49.8 meq/l. [This value includes a constant 1.5 meq/l for sulfate (6) that was not included in the table.] The 49.8 value and the table values of PCO₂ (39.3 Torr) and [Alb] (7.2 g/dl) were used to calculate pH. For filtrand 1, the Fencl model in our hands predicted a pH of 7.443, and the single-association-constant model predicted 7.442. To evaluate the overall fit to the measured pH values, the squared error (S²) was determined using Eq. 6 of Figge, Rossing, and Fencl (6), i.e.

(11)

where n is the number of albumin or serum samples. The significance of the slopes of regression lines was evaluated by t-tests.

	RESULTS

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Fencl Model

View larger version (14K): [in this window] [in a new window]   Fig. 1.   Predictions of albumin model of Figge, Mydosh, and Fencl (5) are compared with pH of albumin solutions measured by Figge, Rossing, and Fencl (6). Slope of least-squares linear fit (not shown) was not different from line of identity (shown).

View larger version (15K): [in this window] [in a new window]   Fig. 2.   Predictions of albumin model of Figge, Mydosh, and Fencl (5) are compared with pH of human sera measured by Figge, Rossing, and Fencl (6). Slope (1.083) of least-squares linear fit (dashed line) was significantly different from 1.0 (P < 0.01).

Single-Association-Constant Model

Determination of fixed charges. With K_H at 1.77 × 10³ eq/l and 16 histidine residues per mole, the variation of S² with the number of fixed charges was determined. There was a clear minimum at 21 eq/mol when S² was 0.0805 (Fig. 3).

View larger version (16K): [in this window] [in a new window]   Fig. 3.   Variation of least-squared error with number of negative fixed charges per mole of albumin. Other model parameters were 16 histidine residues per mole and KH = 1.77 × 107 eq/l.

Albumin data. Figure 4 compares the measured pH (abscissa) with the pH calculated by the single-association-constant model with the parameters as in Fig. 3. The high quality of the fit is evident (S² = 0.0805) and is very similar to the fit of the Fencl model in Fig. 1. The slope of 1.03 was statistically different from 1.0 (P < 0.005).

View larger version (15K): [in this window] [in a new window]   Fig. 4.   Predictions of single-association-constant model are compared with pH of same albumin solutions shown in Fig. 1. Model parameters were 16 histidine residues per mole, 21 fixed negative charges per mole, and KH = 1.77 × 107 eq/l. Slope (1.032) of least-squares linear fit (dashed line) was significantly different from 1.0 (P < 0.05).

3

3

3

3

Human serum data. An excellent fit was observed (S² = 0.088), as shown in Fig. 5. The slope of the regression line (1.10) was similar to the slope found with the Fencl model (1.08) and was significantly different from 1.0 (P < 0.001). The dotted line shows the regression observed when the number of histidine residues per mole was allowed to vary to find the best fit with the net number of fixed charges held at 21 eq/mol. The best fit occurred when the number of histidine residues was 20 eq/mol.

View larger version (15K): [in this window] [in a new window]   Fig. 5.   Predictions of single-association-constant model are compared with pH of same human sera shown in Fig. 2. Slope (1.102) of least-squares linear fit (dashed line) was significantly different from 1.0 (P < 0.01). Dotted line, regression observed when number of histidine residues was 20 eq/mol and net number of fixed charges was 21 eq/mol.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX

The data in Fig. 1 confirm that the detailed work of Figge et al. resulted in an excellent model of the acid-base behavior of human serum albumin. Starting from the number and chemistry of the amino acid components of albumin, they predicted the overall acid-base reactions of this complex molecule over a wide range of conditions. The pH values in Fig. 1 were obtained with PCO₂ of 22.1-91.2 Torr, SID of 22.9-77.9 meq/l, and [Alb] of 1.6-8.0 g/dl, covering most of the clinical ranges. The Fencl model predicted the pH of human serum almost as well as it did albumin (Fig. 2). As argued by Figge, Rossing, and Fencl (6), this implies that serum albumin is by far the most important weak acid in plasma when phosphate is accounted for. Because the predictions are almost entirely based on theory and independently measured parameters, this work solidly supports Stewart's concepts of acid-base balance in body fluids, even if it does not support the exact manner that Stewart applied them. However, the small but significant deviation of the predictions for human serum at the ends of the pH range suggests that a small component of the system is unaccounted for. This might be caused by changes in ion binding with pH, buffering by nonalbumin proteins, or the effects of protein-protein interactions. There is little practical significance to the deviation. From the regression line, it can be calculated that when the actual pH was 7.80, the predicted pH would be 7.83, an error with little clinical significance.

The calculation of the number of fixed charges (Table 1) assumed that all the amino acids in the albumin molecule were able to participate as if they were free in the solution. The fact that the value so obtained (~22 eq/mol) was very close to the value found by fitting argues that the assumption was largely true. The small difference of 1 eq/mol should not be interpreted as indicating that one residue is "hidden," because the process of fitting could vary only the number of fixed charges per mole, and so any other differences in the model that caused a divergence from the data could have been offset by the change from 22 to 21 eq/mol.

The single-association-constant model fits the albumin data almost as well as the Fencl model. Only a close comparison of Fig. 4 with Fig. 1 reveals that some points at the alkaline end of the range lie slightly further above the line of identity than do the Fencl points. This simple model also fits the human serum data closely (Fig. 5). As with the Fencl model, there is a small but significant trend for the model predictions to buffer the changes from pH 7.4 less well than the serum. The slope of the single-association-constant model regression line in Fig. 4 (1.10) was not statistically different from that of the Fencl model in Fig. 2 (1.08). The deviation of the model from the data is not explained by the protein parameters of the model. The net number of fixed charges was already optimized for the albumin data. When the number of histidine residues per mole was allowed to vary to find the best fit to the serum data with the net number of fixed charges held at 21 eq/mol, the best fit occurred when the number of histidine residues was 20 eq/mol (the dotted line in Fig. 5 indicates the regression line). Given the knowledge that albumin contains only 16 histidine residues per mole (5), 20 eq/mol is not a reasonable value, so some unexplained deviation remains, especially at the high pH levels. It seems most likely that the deviation is not caused by the dissociation parameters of the albumin model alone. Again, the practical significance of the deviations of the predictions is small: when the measured pH is 7.80, the single-association-constant model predicts 7.84.

Although the original Stewart model is powerful enough to handle wide ranges of conditions (8, 18), the treatment of the weak acids and the choice of parameter values can be criticized. In contrast, the structure and parameter values of the single-association-constant model were derived from the known properties of serum albumin. This puts the Stewart approach for analyzing blood plasma on a more solid scientific base.

The temperature of the body fluids can vary. Blood temperatures can reach 20°C in the peripheral circulation, and hypothermia is often used in cardiac surgery. Reeves (10) showed how the variation of the histidine pK values with temperature paralleled the corresponding changes in water pK. His concept of the alphastat is an important idea in biology and has direct applications in human physiology (11) and clinical medicine (2, 7). Inasmuch as the most appropriate acid-base management strategy during hypothermia is controversial (2, 7), making the appropriate acid-base calculations easier would contribute to the development of the field. The Stewart approach would be a powerful tool to unravel the chemical complexities, but Stewart's model could not readily be used because of the empirical nature of the treatment of weak acids. As an example of the difficulties, at 37.5°C, Reeves' value of K_A was 1.77 × 10⁷, whereas Stewart used 3.0 × 10⁷. The temperature coefficients for the carbon dioxide and water reactions are well known (3, 10), and the variation of protein buffering with temperature is given by Reeves (10); i.e., the variation of K_H with temperature is known. The present study has shown that when Reeves' K_A value is used at 37.5°C, the single-association-constant model closely fits the albumin and serum data. Hence, by considering the temperature coefficients for the dissociation constants, the single-association-constant model can be used for exploring the effects of temperature on acid-base balance.

It is interesting that pH is much more sensitive to the number of the fixed charges than it is to the value of K_H. Figure 3 shows that increasing or decreasing that number by only 14% (3 eq/mol) more than doubles the total squared error. In contrast, decreasing K_H by 44% or increasing K_H by 70% increases the total squared error by only ~50%. Albumin has many functions, including the binding and transport of electrolytes, metabolites, and hormones. Some of these substances, such as free fatty acids, are charged and carried in millimolar quantities. It is known that pH influences the binding of many substances, but it is also likely that such ligands may influence the acid-base behavior of serum albumin, especially if they alter the number of fixed negative charges. The new model will assist the investigation of these interactions.

The single-association-model combines the scientific core of the Fencl model and the convenience of the Stewart model. To facilitate the use of his model, Stewart provided a solution of the model equations in the form of a fourth-order polynomial (14). The treatment of protein in the single-association-constant model would permit a similar development were it not for the full treatment of the phosphoric acid. Phosphoric acid has three dissociations over a wide pH range. However, inasmuch as two of the dissociations have pK values far removed from the physiological range (pK = 1.9 and 11.8), the error induced by neglecting these is immeasurably small, and phosphate contribution to charge becomes

(12)

Including phosphate, the charge balance equation becomes

(13)

and by substituting for each term by using Eqs. 6, 12, A1, A4, and A5, one obtains

(14)

which can easily be solved numerically if one takes advantage of the fact that the value of the left-hand side rises monotonically from a negative value as [H⁺] rises from 0, becoming positive when the solution value is exceeded. When SID = 38.3 meq/l, PCO₂ = 40 Torr, total phosphorus = 3.6 mg/dl (1.16 mM), and [Alb] = 4.2 g/dl, Eq. 14 gives a pH value of 7.400. In this condition, the total charge on the phosphate ions is 2.1 meq/l, [A_Fixed] = 13.3 meq/l, and [A_H,Tot] = 10.1 meq/l, and the net negative charge on albumin is 11.4 meq/l.

In summary, we have shown that a model of human serum albumin containing a fixed negative charge and a variable positive charge with a pK of the imidazole side chain of histidine can closely predict the pH behavior of human serum albumin and human serum. This structure and the associated parameters can be supported by the literature. The effects of various factors that influence proteins and the physicochemical equilibrium, such as temperature, can be conveniently investigated with this model.