## 3 Results and Discussion
### Kinetic Parameters Determination  
Changing the L-asparagine concentration through the hydrolytic reaction made it possible to investigate the kinetics of free and immobilized ASNase onto pristine and f-MWCNTs. Figure 7 depicts the sigmoidal dependence of the ASNase enzymatic reaction rate on L-asparagine concentration, which is typical of allosteric enzymes. This kinetic behavior is described by the Hill Equation (Equation (1)) [16], and Figure 7 proves the excellent fit of experimental data to this allosteric sigmoidal model. Although [PERSON]'s model for the immobilized enzyme was assumed, in these cases it is important to consider that the mathematic interpretation of reaction-diffusion equations for immobilized catalyzers is much more complex [50]. In fact, the \"apparent constants\" are highly dependent on the working conditions (e.g., the use of rotating electrodes are essential for mass transport, diffusion and solution stirring can modify the real concentration at the enzyme layer, immobilization protocol leads to steric hindrance or claimed multilayers, etc.) [51]. Table 4 summarizes the kinetic properties of free ASNase and both types of ASNase-MWCNT bioconjugate, also presenting the resultant catalytic efficiencies (\(k_{cat}/S_{50}\)). The \(S_{50}\) values for 0.3 and 0.4 mg/mL of free ASNase were 288 and 258 mM, respectively; nevertheless, a reduction in these values was noted when analyzing the immobilized ASNase: 114 and 74 mM, when pristine and f-MWCNTs were used, respectively. The reduction confirmed in \(S_{50}\) for the ASNase-MWCNT complex, similar to the Michaelis-Menten constant (\(K_{M}\)), indicates that the affinity of the enzyme for the substrate improved  
Figure 7: Initial reaction rates (\(v_{0}\)) for both (\(\bullet\)) free and (\(\blacksquare\)) immobilized L-asparaginase (ASNase) onto (**a**) pristine and (**b**) functionalized multi-walled carbon nanotubes (f-MWCNTs-1020) by physical adsorption. The solid lines represent the experimental data fit to the Hill equation. as a consequence of immobilization, hence enhancing its activity. The ASNase particle may be stretched over the MWCNTs surface with better positioning, resulting in increased active site availability and improved affinity for L-asparagine [52]. Through evaluation of \(S_{50}\) for the immobilized enzyme, it was also possible to verify an improved affinity to the substrate when using f-MWCNTs as support. The determination of \(S_{50}\) values also allowed to verify that the use of 189 nm of L-asparagine for the ASNase activity measurement, as reported in the literature [53], was well applied in the case of free ASNase. For the immobilized ASNase lower L-asparagine concentrations could have been used, but it would not allow us to make the intended comparisons. In addition, reported concentration of L-asparagine in leukemia blood serum samples varies from \(10^{-3}\) to \(10^{-2}\) M, while in healthy blood serum samples varies from \(10^{-6}\) to \(10^{-4}\) M [7; 54; 55]. A 2.6-fold reduction on \(v_{max}\) values from 0.044 to 0.017 mM-min\({}^{-1}\) was observed comparing free and immobilized ASNase onto pristine MWCNTs. This decrease was even more pronounced when immobilizing 0.4 mg/mL of ASNase onto f-MWCNTs, with a 4.3-fold decrease in the \(v_{max}\) value. The lower \(v_{max}\) detected after immobilization is due to the diffusion layer around ASNase molecules exhibiting a more significant mass transfer limitation [56], hindering the diffusion of L-asparagine in the direction of ASNase-MWCNTs bioconjugate. The immobilization of enzymes can also decrease molecular flexibility, which is often reflected in an inferior catalytic activity [56] and, subsequently, in a fall in \(v_{max}\) value upon immobilization. When paralleled to the ASNase free form, these kinetic constants resulted in lower catalytic efficiencies (\(k_{cat}/S_{50}\)) of immobilized ASNase (Table 4) due to changes in its three-dimensional structure resulting from the adsorption to a solid support and the consequent steric hindrance.