# Preface

This tutorial was made solely for the purpose of education and it was designed for students taking Applied Math 0340. It is primarily for students who have some experience using Mathematica. If you have never used Mathematica before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA 0330 tutorial. As a friendly reminder, don't forget to clear variables in use and/or the kernel. The Mathematica commands in this tutorial are all written in bold black font, while Mathematica output is in regular fonts.

Finally, you can copy and paste all commands into your Mathematica notebook, change the parameters, and run them because the tutorial is under the terms of the GNU General Public License (GPL). You, as the user, are free to use the scripts for your needs to learn the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately. The tutorial accompanies the textbook Applied Differential Equations. The Primary Course by Vladimir Dobrushkin, CRC Press, 2015; http://www.crcpress.com/product/isbn/9781439851043

Introduction to Linear Algebra

Leonor Michaelis    Maud Menten
In 1903, French-Russian physical chemist Victor Henri found that enzyme reactions were initiated by a binding interaction between the product [P] and the substrate concentration [S]. The relating reaction rate v of formation of product [P] is given by
$v = \frac{{\text d}\,[P]}{{\text d}t} = \frac{V_{\max} [S]}{K_M + [S]} ,$
where Vmax represents the maximum rate achieved by the system, at saturating substrate concentration, and KM is the Michaelis constant. His work was taken up by German biochemist Leonor Michaelis (1875--1949) and Canadian physician Maud Leonora Menten (1879--1960), who investigated the kinetics of an enzymatic reaction mechanism, invertase, that catalyzes the hydrolysis of sucrose into glucose and fructose. In 1913, they proposed a mathematical model of the reaction.
\begin{eqnarray*} \frac{{\text d}\,[E]}{{\text d}t} &=& -k_f [E]\,[S] + k_r [ES] + k_{cat} [ES] , \\ \frac{{\text d}\,[S]}{{\text d}t} &=& -k_f [E]\,[S] + k_r [ES] , \\ \frac{{\text d}\,[ES]}{{\text d}t} &=& k_f [E]\,[S] - k_r [ES] - k_{cat} [ES] , \\ \frac{{\text d}\,[P]}{{\text d}t} &=& k_{cat} [ES] . \end{eqnarray*}
It involves an enzyme, E, binding to a substrate, S, to form a complex, ES, which in turn releases a product, P, regenerating the original enzyme. Here kf is the forward rate, kr is the reverse rate, and kcat is the catalytic rate are all constants.