Automatic determination of ligand purity and apparent dissociation constant (Kapp) in Ca2+/Mg2+ buffer solutions and the Kapp for Ca2+/Mg2+ anion binding in physiological solutions from Ca2+/Mg2+-macroelectrode measurements

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Abstract

Calibration of Ca2+/Mg2+ macroelectrodes and flurochromes in the nmolar and μmolar range, respectively, require the use of buffer solutions. In these buffers the apparent dissociation constant (Kapp) has to be measured since calculation based on tabulated constants gives variable results. The ligand concentration [Ligand]T has also to be estimated. The most accurate and general method for measuring both is the ligand optimisation method based on macroelectrode potential measurements, but this iterative method is time consuming, thus limiting its application. This paper describes an automatic program based on the method, which on entering the measured macroelectrode data calculates Kapp, [Ligand]T and the ionised concentration [X2+] within minutes. This optimisation method cannot be used at Kapp values greater than 0.1 mM, but can be extended into this region if the anion concentration is known. The program has been modified to cover this eventuality. Ca2+/Mg2+ macroelectrodes in conjunction with these programs offer an accurate, routine method for determining Kapp and [Ligand]T in buffer solutions at the appropriate ionic strength, temperature and pH and the Kapp for divalent cations binding to physiological anions under experimental conditions.

Introduction

Ca2+/Mg2+ buffer solutions are routinely used to calibrate macroelectrodes or flurochromes in the nmolar and μmolar range, respectively. Calculation of the [X2+] in these solutions gives variable results depending on the constants used; the calculated [Ca2+] can vary by a factor of 2 and calculated [Mg2+] between 3 and 4.5. Moreover, [Ligand]T has to be independently measured. Because of this, measurement of both the Kapp and the [Ligand]T in the buffers is essential [1]. A review of the methods available for such measurements showed that the most accurate and general method was that of the ligand optimisation method [2]. The procedure is iterative and has the grave disadvantage that it is time consuming—an hour or more being required to evaluate one series of macroelectrode measurements—and this has prevented the widespread use of the method.

In this paper a computer program is described which makes the ligand optimisation method easy to use and enables the calculation of Kapp, [Ligand]T and [X2+] in the buffer solutions from the macroelectrode measurements to be carried out within minutes. We re-formulate the method using a constrained nonlinear least squares approach in which a single objective function of all three parameters is defined and minimised subject to the constraint that the ligand concentration is no greater than its nominal value. The Nicolsky–Eisenman equation relates the measured potentials in the buffer solutions to the values of [X2+]. [X2+] may be written as a function of the [X]T values together with the ligand purity and the apparent equilibrium constant. By substituting this equation for [X2+] into the Nicolsky–Eisenman equation a single nonlinear regression model is obtained which directly links the measured potentials with the corresponding values of [X]T. The associated residual sum-of-squares (RSS) function is then a single objective function which may be directly minimised with respect to all three unknown parameters. This approach is implemented using an Excel spreadsheet which uses the freely available Solver Add-In and contains the program ALE (“Automatic determination of Ligand purity and Equilibrium dissociation constant”), which is driven by macros written in Visual Basic.

There is, however, a limitation to the ligand optimisation method: it may only be applied for values Kapp less than 0.1 mmol/l (pKapp>4). However, if the concentration of the organic anion is known, the method can be easily modified to determine Kapp for divalent cations binding to organic anions of physiological importance (e.g. malate, citrate and aspartate [2], [3]). A modification to the automated method and the program ALE will be presented and illustrated with real data using the program AEC (“Automatic determination of Equilibrium dissociation Constant”).

Both programs have been successfully tested against both real and simulated data. These programs, together with Ca2+/Mg2+ macroelectrodes give a reliable and rapid method to determine not only Kapp and [Ligand]T in Ca2+/Mg2+ buffer solutions but also the Kapp for divalent cation binding to physiological important anions under experimental conditions.

The paper proceeds as follows. In the next section (Section 2), we consider all the technical details used in the program, including the important issue of setting good initial values for the parameters. We discuss the workings of the ALE program in Section 3 and using real data sets, involving Mg2+ and one Ca2+ buffer solutions, we compare the results obtained with ALE and the non-automatic ligand optimisation method. We then apply the ALE program to some simulated data sets in order to check that it produces the correct results. We also discuss issues relating to the reliability of the Excel Solver for nonlinear regression problems in general and discuss results of a small comparative study of the Excel Solver versus the statistical package S-Plus. In Section 4, we describe modifications to the method and the program ALE to deal with the case when Kapp is greater than 0.1 mmol/l and discuss the program AEC in determining Kapp for divalent cations binding to an organic anion of known purity. Using real data sets, we compare the results obtained with ALE and the original non-automatic method of calculation and similarly for AEC. Finally, in Section 5, we present our conclusions. The work has been published in abstract form [1], [4]. The definitions of the scientific abbreviations used in the paper are given in Appendix A.

Section snippets

The program ALE

The relative potentials in the buffer solutions together with the characteristics of the macroelectrode formed the basis for the ligand optimisation method [5]. The program ALE is embedded within a spreadsheet interface which allows the user to enter the data, press the program button ALE and read off the required answers within minutes. The spreadsheet is shown in Fig. 1 for the data shown in Table 1.The calibration data are entered, namely [Ca]T in each calibration solution, relative time and

Program AEC for estimation of Kapp of Mg2+ and Ca2+ binding to anions

The ligand optimisation method [5] has one clear limitation: at values of Kapp>0.1mmol/l(pKapp<4) the method breaks down because the term Kapp+[Ligand]T-[X]Tin Eq. (3) becomes dominated by Kapp and so is insensitive to changes in [Ligand]T. Consequently, it is not possible to obtain precise results for the [Ligand]T parameter when optimising the objective function in Eq. (7). This limitation is not a problem with calcium and magnesium buffer solutions which have pKapp values greater than 5.5,

Conclusions

This paper describes an automated method for implementing the ligand optimisation method of [5] and it was demonstrated, by means of experimental data sets, a small simulation study and by comparison with a professional statistical package, that the program ALE produces accurate results. The program is available via an MS Excel spreadsheet and is very easy to use: simply enter the data, check the solver constraint and press the ALE program button. A direct comparison between the non-automated

Summary

Ca2+/Mg2+ buffer solutions are routinely used to calibrate macroelectrodes or flurochromes in the nmolar and μmolar range, respectively. Calculation of the [X2+] in these solutions gives variable results depending on the constants used; the calculated [Ca2+] can vary by a factor of 2 and calculated [Mg2+] from between 3 and 4.5. Moreover, [Ligand]T has to be independently measured. Because of this, measurement of both the Kapp and the [Ligand]T in the buffers is essential. The best

Jim Kay was until recently a senior lecturer in Statistics at the University of Glasgow. He holds the degrees of B.Sc. (1st class Hons.) in Mathematics and a Ph.D. in Statistics from the University of Glasgow as well as the degree of BD from the University of Edinburgh. He is co-editor (with D.M. Titterington) of the book Statistics and Neural Networks: Advances at the Interface and a co-author (with J. Aitchison and I.J. Lauder) of the book Statistical Concepts and Applications in Clinical

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Jim Kay was until recently a senior lecturer in Statistics at the University of Glasgow. He holds the degrees of B.Sc. (1st class Hons.) in Mathematics and a Ph.D. in Statistics from the University of Glasgow as well as the degree of BD from the University of Edinburgh. He is co-editor (with D.M. Titterington) of the book Statistics and Neural Networks: Advances at the Interface and a co-author (with J. Aitchison and I.J. Lauder) of the book Statistical Concepts and Applications in Clinical Medicine. He has authored or co-authored more than 50 refereed articles in statistical, engineering and scientific journals.

Rachel Steven's interests in sport, physiology and computing led to her graduating with an honours degree in Sports Physiology (2001) from the University of Glasgow. She is a past President of Glasgow University Sports Association. Her interest in computational data manipulation and analysis led to her project, developing an early version of the program described here, in conjunction with the other authors. She has now left academia for a career in financial services.

John McGuigan combined the study of medicine at the Glasgow University, with the study of physiology, graduating in 1961. After his year in hospital, he completed his Ph.D. in physiology in Glasgow and in 1967 joined the Institute of Physiology, University of Bern, Switzerland, where Professor Silvio Weidmann was chair. He retired from Bern in 1999 as titular professor and since then have been Senior Honorary Research Fellow IBLS, at Glasgow University. His main research interest has been Mg2+ regulation in heart, but a side line to this was the accurate manufacture of calcium and magnesium buffer solutions. He is a member of the Physiological Society and was chair of the Gordon Research Conference on “Magnesium in Medicine and Biological Processes” in 1999.

Hugh Elder graduated initially from Glasgow University with a degree in Marine Zoology (1st class Hons.) in 1959. During undergraduate studies histological training under HF Steedman and HSD Garven engendered a love of functional microanatomy and microscopical techniques. Following a brief career in fisheries biology, including some years in East Africa, his interests in functional morphology in Ph.D. studies led to a move into physiology and interest in ion transport and secretory processes. Amongst many techniques employed, light and electron microscopy, X-ray microanalysis and microspectrofluorimetry he continued his microscopical interests and he is a past Hon. Secretary for Science, and past President of the Royal Microscopical Society. Author or co-author of over a hundred papers, he retired as Reader in Physiology in 2001 and remained as an Honorary Research Fellow of IBLS in the University of Glasgow.

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