Imidazole binding to met-haemoglobin — a convenient experiment for the illustration of multiple equilibria

44

Experimental Section

Imidazole Binding to Met-Haemoglobin - - a Convenient Experiment for the Illustration of Multiple Equilibria T BRITTAIN

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Introduction In most biochemistry courses the topic of multiple equilibria and its analysis is often covered in lectures. Most commonly the material covers (i) single sites, (ii) multiple independent sites, of more than one type, and (iii) multiple interacting sites, and exposes the students to the principles of such analytical approaches as Scatchard analysis and logit plots. Unfortunately very few, easily amenable, laboratory exercises are available to illustrate these topics. Experimental exercises which do allow students to obtain 'real data' often require the handling of radioactive materials or else involve impractically prolonged periods of waiting, as in the case of equilibrium dialysis studies. In order to alleviate these problems whilst still allowing students to obtain a direct appreciation of the strengths and weaknesses of the various theoretical approaches covered in lectures, we have developed a laboratory exercise designed to illustrate the experimental determination of equilibrium constants for a system of multiple independent sites which is both rapid and convenient. Furthermore the experiments have been designed to allow students to quickly obtain very precise data suitable for subsequent analysis by a variety of methods aimed at highlighting the advantages and limitations of each method using their own 'real data'.

Approach The experimental approach employed in the practical illustration of multiple equilibria centres on the spectrophotometric measurement of imidazole binding to met-haemoglobin. Although the binding of imidazole to met-haemoglobin was formerly considered to be a simple process I clear evidence now exists that the binding process consists of two independent binding processes 23 ' which can be identified and quantified using data obtained from spectrophotometric titration. Met-haemoglobin binds imidazole with a low affinity; a useful property in that the effects of student errors in measuring both haemoglobin and ligand concentrations are minimised. The binding of imidazole to met-haemoglobin is accompanied by large changes in the absorption spectrum in the 500-600 nm region (see Fig 1). These changes are easily monitored using single wavelength spectrophotometry and the low affinity for ligand means that the concentration of met-haemoglobin used in the experiment can be quite high, without affecting the hyperbolic character of the binding process, 4 and so can be adjusted accordingly to suit the sensitivity and stability of the spectrophotometer employed. Thus the student performs a simple, single, spectrophotometric titration of met-haemoglobin with imidazole. Depending on the time available it is quite possible for the students to prepare their own met-haemoglobin by ferricyanide oxidation, followed by chromatography on a Sephadex G25 column. If one or more recording spectrophotometers are available the students can also obtain spectra of met-haemoglobin and met-haemoglobin in the presence of saturating amounts of imidazole and hence determine the most appropriate wavelength at which to follow the binding process.

BIOCHEMICAL EDUCATION 18(1) 1990

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Figure 1 Absorption spectra of 50 p.M met-haemoglobin ( ) and the corresponding met-haemoglobin-imidazole complex (--) at p H 7.2 in 0.05 M phosphate buffer

Procedure and Results The students obtain a set of A56onm versus [Im]Total data by titrating 3 ml of 50 pxM met-haemoglobin with aliquots of imidazole solutions. All solutions are prepared in 0.05 M phosphate buffer at pH 7.2. It is necessary to use imidazole solutions of 4 M, 1 M and 0.1 M in order to obtain the concentration range required and these solutions are titrated to pH 7.2 with HCI prior to their use in the titration procedure. Each addition of imidazole should be followed by a 1-2 min wait to allow equilibration prior to the recording of A560. The titration is most easily and accurately performed using a spectrophotometer with a digital display. If only relatively low sensitivity spectrophotometers are available the protein concentration can be raised to 100 IxM with little effect other than raising the observed absorbance change. The volumes added at each stage are adjusted from a few microliters of the 0.1 M imidazole solution at the beginning of the titration to 50-100 txl of the 4 M imidazole solution at the end of the titration, the volume of addition at each stage being chosen so as to produce approximately equal absorbance changes at each step. Approximately 15-20 additions produce a usable data set corresponding to a final concentration of - 1 5 0 mM imidazole. Ligand concentrations significantly above this level should not be employed as they lead to slow irreversible changes in the protein spectrum. The students are then faced with the problems of analysis of this 'real data' set. The analysis proceeds in a series of single steps which are designed to illustrate the various strengths and weaknesses of each form of analysis. Firstly, the students are reminded that all titrations are asymptotic to their end-point and that a good-end point value is essential for detailed analysis. The students produce a CornishBowden type direct linear plot of Asa~ versus [Im] to obtain a value for the A560 at infinite Jim] (Fig 2). At this point they should identify that the last few data points yield a good intersection point but that the first data points do not yield the same intercept, and they have to appreciate that this finding might represent error or the presence of multiple binding sites. Using the A560 final value the students next tabulate their results in terms of AA56o (corrected for dilution); fractional saturation, Y, given by ~d~obs/~Z~total and Jim]free using [Im]free = [Im]tota] - [Hb]Y. They are reminded that this analysis for Y is based on the axiom that AA is linearly related to Y, and then produce a Hill Plot of their data (Fig 3). This plot is used to identify the binding process as either arising from the presence of negative co-operativity or binding to multiple sites.

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Figure 2 Direct linear plot of titration data of imidazole binding to met-haemoglobin obtained at 560 nm It can also be used to illustrate the fact that there is a tendency to draw the asymptotes of unit slope too close to the experimental line (cf lines drawn in Fig 3 obtained from more detailed analysis) yielding an underestimate of the high affinity binding constant and an overestimate of the low affinity binding constant. Given that haemoglobin has four binding sites the next step is to derive a set of values of v (using v = 4Y/[Hb]) and v/[Im] and produce a Scatchard Plot of the data (Fig 4). This plot is used to illustrate the point that, except for the intercept on the v axis, the values of the intercepts and slopes of the limiting lines, obtained using the parameters of the two site model, do not take the values which might intuitively be expected for binding to two different types of site. This complexity of Scatchard Plots, for multiple equilibria, is, in our

Figure 4 Scatchard Plot of met-haemoglobin binding of imidazole. ~ represents the average number of bound ligand molecules per haemoglobin molecule. [X] represents [Im]/.... The two lines are calculated from the best fit two site data shown in Fig 5b case, covered in complementary lecture material which deals with the rigorous analysis of Klotz and Hunston. 5 At this stage of the analysis the students should have a clear appreciation that their original data set probably represents the binding of imidazole to two sets of sites, and that their Hill Plots and Scatchard Plots give only relatively poorly defined binding constants. It is at this point that we progress to the less commonly described computer fitting approach. The students enter their data in the form of Y versus [Im]tr¢¢ into two BASIC programs run on PCs. Both programs perform non-linear least squares fitting of the data set but to different binding equations. 3 Firstly the data are fitted to Y = K[Imlf~ed(1 + K[Imlfree) as shown in Fig 5a. The students input an initial guess for K which can simply be obtained from the [Im]free corresponding to Y = 0.5, and obtain a best fit value, a comparison of Yobs versus Yc~dc, a graph of the fit and a graph of the distribution of the residuals, ie the difference between the observed Y values and the Y value calculated from the model at each [Ira]free, together with a value for the residual sum of the squares of the deviations. The residual sum of the squares indicates a moderate fit but the graph of the distribution of residuals shows large systematic deviations which represent a clear indication of the inappropriateness of this model. The students next fit their data to

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BIOCHEMICAL EDUCATION 18(1) 1990

and are presented with similar output to that described above. In this case the students can introduce a value of K 1 obtained from their direct linear plot and a value of K2 of K1/10 for their initial estimates of the binding constants. The residual sum of squares is much improved and, although our students are not adequately prepared for a discussion of the statistical significance of this improved fit with regards to the increased number of degrees of freedom, the graph of the distribution of the residuals (Fig 5b) which clearly indicates smaller residuals with a random distribution, is taken as an indicator of the appropriateness of the two binding site model. The best fit values of K1 and K2 can then be used to derive values of the intercepts for the Hill Plot and Scatchard data, and clearly demonstrate the limitations of these methods (see Fig 3 and 4).

46 material, this laboratory significantly enhances students appreciation of both the practical and theoretical aspects of the analysis of systems showing multiple equilibria. References

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tBeetlestone, J G, Epega, A A and lrvine, D H (1968) J Chem Soc 6, 1346-1351 2Uchida, H, Heystek, J and Klapper, M H (1971) J Biol Chem 246, 2031-2034 3Brittain, T (1981) J lnorg Biochem 15, 243-252 4Van Holde, K (1985) 'Physical Biochemistry', Second Edition. Prentice-Hall, USA 5Klotz, I M and Hunston, D k (1971) Biochemistry 10, 3065-3069

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Figure 5 A comparison of experimental ( . . . ) and theoretical data ( ) for the binding of imidazole to met-haemoglobin using a single site model (a) and a two site model (b). The best fit to (a) gave the value of K = 210 M -1 and a residual sum of squares of 2.1 x 10 -2. The best fit to (b) gave values of K1 = 80 M -~ and K2 = 425 M -I with a residual sum of the squares of the deviations of 1.1 x 10 -3. (The best fit values of (b) were used to draw the straight lines in Figs 3 and 4). R represents the quantity (Y observed - Y calculated), where Y calculated is the value of Y calculated from the best fit K value and the appropriate [Im]fr~e

Introduction

Practical courses in biochemistry usually include experiments where use is made of recorders to obtain a visual and quantitative evaluation of a particular process. If the apparatus is not equipped with an integrator or microprocessor, quantitative evaluation of recorder graphs may be difficult for students not proficient in mathematics. Typical examples are the determination of the change in some process (absorbance, oxygen uptake, temperature, etc) with time by calculating the gradients of the graphs obtained during the experiment. The time-consuming method of counting graph squares for determining gradients may be avoided by using the following method which is easily applied using a protractor and a simple formula. Protractor Method

The formula is derived as follows. Consider a process (Y) which changes linearly with time (X). The process may decline (Figure 1A) or increase (Figure 1B) with time. The gradient of such a graph may be calculated by applying the following formula: gradient = C × cotangent T

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Experimental Computation

Computations were performed in BASIC on a PC equipped with a Colour Graphics Adapter (CGA) monitor. To enhance the speed of the calculations the programs were run in compiled form using Quick BASIC. Both programs are available free from the author on request (send a blank disk). Conclusions

The binding of imidazole to the ~ and 13 chains 3 of methaemoglobin represents an excellent vehicle for the demonstration of multiple equilibria to undergraduate students. The low affinity of the protein for imidazole allows a wide range of experimental conditions to be employed and which can be optimised according to the spectrophotometric equipment available. The similarity of the two binding constants permits the illustration of various forms of analysis and highlights their limitations. The use of this spectrophotometric approach allows students to obtain the degree of accuracy necessary in their data for subsequent analysis. Furthermore the data are obtained in a relatively short time and do not necessitate the use of radioactive materials. Particularly when integrated with appropriate lecture

BIOCHEMICAL EDUCATION 18(1) 1990

where T = angle as indicated in Fig 1 and determined by using a protractor, and C = constant calculated as shown below. The constant (C) for a particular set of graphs need be calculated only once for a given graph paper and paper speed, and can be keyed into a pocket calculator. For the normal situation where say 25 squares (p millimetres) on the X-axis represents Xl time units and say 10 squares (q millimetres) on the Y-axis represents Yl units of the process investigated, the value for C will be given by C = pyl/xlq

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where p = scale of X-axis (25 squares represent p millimetres), q = scale of Y-axis (10 squares represent q millimetres), xj = unit scale of X-axis (25 squares represent xl seconds), and Yl = unit scale of Y-axis (10 squares represent Yl units). The values for xl and Yl will depend on the adjustments made on the apparatus and recorder, while p and q will depend on the graph paper used. The given formula (Eqn 2) for C will only be valid for cases where x~ and p are taken over the same amount of