Topic 8 Appendix A: Calculation of Equilibrium Constant for the Binding of Imidazole to Ferrihemoglobin

Russell Larsen; Mouna Maalouf

Topic 8 Appendix A: Calculation of Equilibrium Constant for the Binding of Imidazole to Ferrihemoglobin

In contrast to the complex oxygen binding properties of ferrohemoglobin, in which the four subunits of protein act in a cooperative manner, the binding of ligands by ferrihemoglobin is often chemically simpler. In this case, binding can often be explained by assuming that each subunit binds ligands independently. In other words, the overall equilibrium has a single equilibrium constant. If Hb is taken to stand for a single subunit of the hemoglobin protein, the imidazole binding of ferrihemoglobin can be described by the following reaction:

Hb•Fe⁺_(aq) + Im_(aq) $\right)\rightleftarrows$ Hb•Fe–Im⁺_(aq),

in which Hb•Fe⁺ represents ferrihemoglobin with water bound to the heme iron, Im represents imidazole, and Hb•Fe – Im⁺ represents the ferrihemoglobin-imidazole adduct.
At constant pH, the equilibrium constant for the reaction is given by:

$\displaystyle Keq=\frac{{{[Hb•Fe-Im^{+}]}}}{{[Hb•Fe^{+}]}{[Im]}}}$ .

In order to determine K_eq for imidazole binding to ferrihemoglobin, the concentrations of the [Hb•Fe–Im⁺], [Im], and [Hb•Fe⁺] will be measured. Excess imidazole is used in these experiments so that [Im]_initial ≈ [Im]_eq. The spectrophotometer can be used to measure the concentrations of Hb•Fe⁺ and Hb•Fe–Im⁺ since both species absorb light in the visible region of the spectrum. The analysis is complicated by the fact that the reactant, Hb•Fe⁺, and the product, Hb•Fe–Im⁺, both absorb light at visible wavelengths. However, as long as the two forms have different spectral analysis, the binding constant can be obtained by a systematic study of the absorbance as a function of the imidazole concentration.

To begin the analysis, assume that at a particular wavelength of light the absorbance of a given concentration of Hb•Fe⁺ is A₀ (i.e. the absorbance with 0% of the hemoglobin bound to imidazole). At this same wavelength of light and protein concentration, the absorbance of Hb•Fe-Im⁺ is A₁₀₀ (i.e. the absorbance with 100% of the hemoglobin bound to imidazole). The resulting expression for K_eq is shown below

$\displaystyle Keq=\frac{{{[Hb•Fe-Im^{+}]}}}{{[Hb•Fe^{+}]}{[Im]}}}$ $\displaystyle =\frac{{{1}}}{{[Im]}}$ * $\displaystyle \frac{{{(A-A_{0})}}}{{(A}_{100}{-}{A})}}$ .

where A is the absorbance measured at equilibrium in an experimental solution, A₀ is the absorbance of all of the hemoglobin in the Hb•Fe⁺ state, and A₁₀₀ is the absorbance of all of the hemoglobin subunits in the Hb•Fe – Im⁺ state. Although absorbance of the unbound form, A₀, is easily experimentally accessible, the form of the completely imidazole-bound form, A₁₀₀, is only accessible in as an asymptotic limit. For this reason, it is useful to identify a useful mathematic form of this equation to use to simplify the analysis. By inserting zero in the clever form of (A₀ – A₀) into the denominator of the above expression, a more useful form can be obtained:

$\displaystyle Keq=\frac{{{[Hb•Fe-Im^{+}]}}}{{[Hb•Fe^{+}]}{[Im]}}}$ $\displaystyle =\frac{{{1}}}{{[Im]}}$ * $\displaystyle \frac{{{(A-A_{0})}}}{{(A}_{100}{-}{A})}}$ $\displaystyle =\frac{{{1}}}{{[Im]}}$ * $\displaystyle \frac{{{(A-A_{0})}}}{{(A}_{100}{-}{A} {+}{A}_{0}{-}{A}_{0})}}$

$\displaystyle =\frac{{{1}}}{{[Im]}}$ * $\displaystyle \frac{{{(A-A_{0})}}}{{(A}_{100}{-}{A}_{0}){-}({A}{-}{A}_{0})}}$ .

This equation can be further manipulated by inverting both sides to obtain:

$\displaystyle \frac{{{1}}}{{Keq}}$ $\displaystyle =[Im]$ $\displaystyle\left( \displaystyle \frac{{{(A_{100}-A_{0})}}}{{(A}{-}{A}_{0})}} - \displaystyle \frac{{{(A-A_{0})}}}{{(A}{-}{A}_{0})}}\right)$ $\displaystyle =[Im]$ $\displaystyle \left(\displaystyle\frac{{{(A_{100}-A_{0})}}}{{(A}{-}{A}_{0})}} - \displaystyle {1}\right)$

which can be rearranged to,

$\displaystyle \frac{{{1}}}{{Keq}}$ $\displaystyle *\frac{{{1}}}{{[Im]}}$ + $\displaystyle {1}$ $\displaystyle =\left(\frac{{{(A_{100}-A_{0})}}}{{(A}{-}{A}_{0})}}\right)$

and further,

$\displaystyle \frac{{{1}}}{{(A_{100}-A_{0})}{Keq}}$ $\displaystyle *\frac{{{1}}}{{[Im]}}$ + $\displaystyle \frac{{{1}}}{{(A_{100}-A_{0})}}$ $\displaystyle =\left(\frac{{{1}}}{{(A}{-}{A}_{0})}}\right)$

At this point, it is important to recognize that it has the form of a straight line, y=mx+b.

$\displaystyle \frac{{{1}}}{{(A_{100}-A_{0})}{Keq}}$ $\displaystyle *\frac{{{1}}}{{[Im]}}$ + $\displaystyle \frac{{{1}}}{{(A_{100}-A_{0})}}$ $\displaystyle =\left(\frac{{{1}}}{{(A}{-}{A}_{0})}}\right)$

if we define ΔA = A – A₀, then this gives

$\displaystyle \frac{{{1}}}{{\Delta}{A}_{100}{Keq}}}$ $\displaystyle *\frac{{{1}}}{{[Im]}}$ + $\displaystyle \frac{{{1}}}{{\Delta}{A}_{100}}$ $\displaystyle =\left(\frac{{{1}}}{{\Delta}{A}}\right)$

which corresponds to mx+b = y with

m = $\displaystyle \frac{{1}}{{\Delta}{A}_{100}{Keq}}$ , b = $\displaystyle \frac{{1}}{{\Delta}{A}_{100}}$ , x = $\displaystyle \frac{{1}}{[Im]}}$ , and y = $\displaystyle \frac{{1}}{{\Delta}{A}}$

Thus, the equilibrium constant can be obtained from a best fit line simply as,

K_eq = $\displaystyle \frac {b}{{{m}}}$ $\displaystyle =\frac {intercept}{{slope}}$ .

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