Automatic determination of ligand purity and apparent dissociation constant in buffer solutions and the for anion binding in physiological solutions from -macroelectrode measurements
Introduction
buffer solutions are routinely used to calibrate macroelectrodes or flurochromes in the nmolar and molar range, respectively. Calculation of the in these solutions gives variable results depending on the constants used; the calculated [] can vary by a factor of 2 and calculated between 3 and 4.5. Moreover, has to be independently measured. Because of this, measurement of both the and the 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 , and 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 . may be written as a function of the values together with the ligand purity and the apparent equilibrium constant. By substituting this equation for into the Nicolsky–Eisenman equation a single nonlinear regression model is obtained which directly links the measured potentials with the corresponding values of . 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 less than 0.1 mmol/l . However, if the concentration of the organic anion is known, the method can be easily modified to determine 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 macroelectrodes give a reliable and rapid method to determine not only and in buffer solutions but also the 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 and one 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 is greater than 0.1 mmol/l and discuss the program AEC in determining 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 in each calibration solution, relative time and
Program AEC for estimation of of and binding to anions
The ligand optimisation method [5] has one clear limitation: at values of the method breaks down because the term in Eq. (3) becomes dominated by and so is insensitive to changes in . Consequently, it is not possible to obtain precise results for the parameter when optimising the objective function in Eq. (7). This limitation is not a problem with calcium and magnesium buffer solutions which have 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
buffer solutions are routinely used to calibrate macroelectrodes or flurochromes in the nmolar and molar range, respectively. Calculation of the in these solutions gives variable results depending on the constants used; the calculated [] can vary by a factor of 2 and calculated [] from between 3 and 4.5. Moreover, has to be independently measured. Because of this, measurement of both the and the 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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Cited by (10)
Ionised concentrations in calcium and magnesium buffers: Standards and precise measurement are mandatory
2017, Progress in Biophysics and Molecular BiologyIonised concentrations in calcium and magnesium buffers: Standards and precise measurement are mandatory
2016, Progress in Biophysics and Molecular BiologyCitation Excerpt :Since the method has been fully described in previous publications, Lüthi et al. (1997), McGuigan et al. (2006), Kay et al. (2008), McGuigan and Stumpff (2013) and McGuigan et al. (2014), it will not be described here. The method has also been verified by; a), measurement of the total [EGTA] by CaCl2 titration, was indistinguishable from calculation with LOM (Fig. 6B, McGuigan et al., 2006); b) calibration of a Ca2+-electrode in EGTA, BAPTA, HEDTA and NTA gave overlapping and contiguous results on a single calibration curve calculated by LOM (McGuigan and Stumpff, 2013); c) calibration of a Mg2+-electrode by dilution alone from 10 μmol/l to 10 mmol/l was identical with results calculated with LOM (Fig. 3B, Lüthi et al., 1997); d) Ligand purity and K/ were correctly predicted with the use of simulated data sets (Kay et al., 2008; McGuigan et al., 2014; JAS McGuigan, unpublished). LOM is regarded as the method of choice to measure the [X2+] in Ca2+/Mg2+ buffer solutions.
Measuring Ca<sup>2+</sup> binding to short chain fatty acids and gluconate with a Ca<sup>2+</sup> electrode: Role of the reference electrode
2014, Analytical BiochemistryCitation Excerpt :It is this apparent constant that is measured in these experiments, which should be independent of the concentration of the ion under investigation (see Appendix to Ref. [10]). Unless indicated otherwise, the Kapp was calculated assuming one-to-one binding between Ca2+ and the anionic ligand using the Excel program AEC [11], which yields a fit to the potentials measured by a Ca2+ macroelectrode to the general form of the Nikolsky–Eisenman equation. Data were tested for normal distribution using the Kolmogorov–Smirnov test.
An improvement to the ligand optimisation method (LOM) for measuring the apparent dissociation constant and ligand purity in Ca<sup>2+</sup> and Mg<sup>2+</sup> buffer solutions
2014, Progress in Biophysics and Molecular BiologyCitation Excerpt :The data sets are taken from Fig. 2A (Mg2+-EDTA) and Fig. 4B (Ca2+-EGTA) from McGuigan et al. (2007). This method is described in Lüthi et al. (1997), McGuigan et al. (2006) and Kay et al. (2008). The method consists of 4 steps and the first three are illustrated in Fig. 1 based on an Ca2+-EGTA experiment at 25 °C and a pHa of 7.2.
Calculated and measured [Ca<sup>2+</sup>] in buffers used to calibrate Ca<sup>2+</sup> macroelectrodes
2013, Analytical BiochemistryIonized concentrations in Ca<sup>2+</sup> and Mg<sup>2+</sup> buffers must be measured, not calculated
2020, Experimental Physiology
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 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.