Progress-Curve Analysis Through Integrated Rate Equations and Its Use to Study Cholinesterase Reaction Dynamics

Abstract

Michaelis and Menten found the direct mathematical analysis of their studied enzyme-catalyzed reaction unrealistic 100 years ago, and hence, they avoided this problem by correct adaptation and analysis of the experiment, i.e., differentiation of the progress-curve data into rates. However, the most elegant and ideal simplification of the evaluation of kinetics parameters from progress curves can be performed when the algebraic integration of the rate equation results in an explicit mathematical equation that describes the dynamics of the model system of the reaction. Recently, it was demonstrated that such an alternative approach can be considered for enzymes that obey the generalized Michaelis–Menten reaction dynamics, although its use is now still limited for cholinesterases, which show kinetics that deviate from saturation-like hyperbolic behavior at high concentrations of charged substrates. However, a mathematical approach is reviewed here that might provide an alternative to the decades-old problem of data analysis of cholinesterase-catalyzed reactions, through the more complex Webb integrated rate equation.

Appendix

Software-user-defined, built-in equations using fourth-order term approximations of solutions for product concentrations in GraphPad Prism 5.

s:

=S0/Kss

R:

=Ks/Kss

t:

=Vm*x/Kss

sR:

=(s + R)*(1 + s)

f0:

=s*(1 + b*s)/sR

f1:

=((−1 + b)*s^2 + R*(1 + b*s*(2 + s)))/sR^2

f2:

=−(2*(−(−1 + b)*R^2 + (−1 + b)*s^3 + R*(1 + 3*s + 3*b*s^2 + b*s^3)))/sR^3

f3:

=6*((1 − b)*R^3 + (b − 1)*s^4 − (b − 1)*R^2*(1 + 4*s) + R*(1 + 4*s + 6*s^2 + 4*b*s^3 + b*s^4))/sR^4

s1:

=−t*f0

s2:

=−(t/2)*s1*f1

s3:

=−(t/3)*(f1*s2 + f2*s1^2/2)

s4:

=−(t/4)*(f1*s3 + f2*s1*s2 + f3*s1^3/6)

Y:

=−Kss*(s1 + s2 + s3 + s4)