## 6 Mechanistic Details of Autoxidation of Human HbO\({}_{2}\): Computer Simulations  
To shed light on the mechanistic details of human HbO\({}_{2}\) autoxidation, we here propose two tentative models and make a kinetic formulation. Curve fittings were made against the data obtained by UV/VIS spectrophotometric, isoelectric-focusing electrophoretic and polyacrylamide gel electrophoretic measurements. According to our specifications [37] to solve differential equations numerically using the Runge-Kutta method by setting a tentative reaction model, concentration progress curves for species assumed in a given model can be displayed on a personal computer at a given time. The first tentative model can be written as follows:  
In this scheme, it is assumed that the reaction can proceed via two valency hybrid intermediates, _i.e._, (\(\alpha^{3+}\beta^{2+}\))\({}_{2}\) and (\(\alpha^{3+}\beta^{2+}\))\({}_{2}\). According to the differential equations deduced from the reaction model shown below, we formulated possible pathways leading to the oxidation of Hb to metHb via two valency hybrid intermediates and drew up all the concentration progress curves for the species involved in this reaction. \[\mathrm{d}[(\alpha^{2+}\beta^{2+})_{2}]/\mathrm{dt}=-(k_{1}+k_{2}) \times[(\alpha^{2+}\beta^{2+})_{2}] \tag{8}\] \[\mathrm{d}[(\alpha^{3+}\beta^{2+})_{2}]/\mathrm{dt}=k_{1}\times[( \alpha^{2+}\beta^{2+})_{2}]-k_{3}\times[(\alpha^{3+}\beta^{2+})_{2}]\] (9) \[\mathrm{d}[(\alpha^{2+}\beta^{3+})_{2}]/\mathrm{dt}=k_{2}\times[( \alpha^{2+}\beta^{2+})_{2}]-k_{4}\times[(\alpha^{2+}\beta^{3+})_{2}] \tag{10}\]  
and  
\[[(\alpha^{3+}\beta^{3+})_{2}]_{t}=[(\alpha^{2+}\beta^{2+})_{2}]_{t=0}-[( \alpha^{2+}\beta^{2+})_{2}]_{t}-[(\alpha^{3+}\beta^{2+})_{2}]_{t}-[(\alpha^{2+ }\beta^{3+})_{2}]_{t} \tag{11}\]  
Here the total concentration of Hb (= \([(\alpha^{2+}\beta^{2+})_{2}]_{t=0}\)) is given by the following equation:  
\[[(\alpha^{2+}\beta^{2+})_{2}]_{t=0}=[(\alpha^{2+}\beta^{2+})_{2}]_{t}+[(\alpha ^{3+}\beta^{2+})_{2}]_{t}+[(\alpha^{2+}\beta^{3+})_{2}]_{t}+[(\alpha^{3+}\beta ^{3+})_{2}]_{t} \tag{12}\]  
The numerical successive approximation was achieved using the Runge-Kutta method by stepping up one-by-one along with the steps shown in the Appendix. Figure 12(a) illustrates the best-suited fitting for the consecutive changes of the components attained by isoelectric-focusing electrophoresis while setting \(\Delta t\) = 0.1 h. In Figure 12(a), the first (\(\circ\); \([\alpha^{2+}\beta^{2+}]_{2}\)), second (\(\Delta\); \([\alpha^{3+}\beta^{2+}]_{2}\)), third (\(\Delta\); \([\alpha^{2+}\beta^{3+}]_{2}\)) and fourth (\(\bullet\); \([\alpha^{3+}\beta^{3+}]_{2}\)) components were plotted against time in terms of peak area calculated from the diagrams shown in Figure 8. Each solid line was achieved by non-linear least-squares fitting when setting the kinetic constants as follows: \(k_{1}\) = 0.009 (h\({}^{-1}\)), \(k_{2}\) = 0.002 (h\({}^{-1}\)), \(k_{3}\) = 0.005 (h\({}^{-1}\)), and \(k_{4}\) = 0.00008 (h\({}^{-1}\)). The resulting curve fittingsshow mutually good consistency for the aspects resulting from the isoelectric-focusing electrophoretic study. However, it was evident that the obtained concentration progress curves derived from this model did not fully explain the aspects of spectrophotometric and polyacrylamide gel electrophoretic profiles. Figure 12(b) shows such an example. Here \(k_{\rm A}^{\,f}\) and \(k_{\rm A}^{\,s}\) represent the first-order rate constants calculated from the least-squares fittings against -ln([HbO\({}_{2}\)]/[HbO\({}_{2}\)]\({}_{0}\)) _versus_ time plot that was monitored in 0.05 M phosphate buffer (pH 5.0) at 37 \({}^{\circ}\)C in the presence of 20% (_v/v_) glycerol (Figure 7(a)). By these least-squares fittings, the following kinetic constants and molar fraction of the initial fast component (_P_) were established:  
Figure 13: Outcomes achieved by a computer simulation using the consecutive reaction model with \(k_{\rm A}^{\,f}\) and \(k_{\rm A}^{\,s}\) for human HbO\({}_{2}\) autoxidation.