May 21, 2018, Monday

# Analyzing the Closed Michaelis-Menten Model

## Introduction

The Michaelis-Menten model is the simplest model for substrate-enzyme kinetics and is classically derived so as to relate reaction rate to the concentration of substrate. In the model, the substrate $S$ binds in a reversible reaction to the enzyme $E$, forming an intermediate complex $C$, which breaks down in a typically non-reversible reaction to form the product $P$:

$S+E{\xleftarrow {v_{{-1}}}}{\xrightarrow {v_{1}}}C{\xrightarrow {v_{2}}}P+E$

## Derivation

To obtain an analytical expression for the velocity of the closed Michaelis-Menten reaction, the Law of Mass Action must first to applied to obtain a system of four differential equations for the substrate, enzyme, complex, and product:

${\begin{cases}{\frac {dS}{dt}}=k_{{-1}}C+k_{1}SE\\{\frac {dE}{dt}}=(k_{{-1}}+k_{2})C-k_{1}SE\\{\frac {dC}{dt}}=k_{1}SE-(k_{{-1}}+k_{2})C\\{\frac {dP}{dt}}=k_{2}C\end{cases}}$

An expression for the velocity of the reaction, or the rate at which the product is produced, cannot be obtained without some sort of analytical expression for the reactants, specifically that of the complex. Thus, analysis of this system typically proceeds by taking advantage of the conservation relationship $E+C=E_{0}$, which can be found using the left null space, a systems biology technique discussed here, allowing for a reduction in dimension of the system. Then, the quasi-steady-state assumption can be applied: a characteristic of substrate-enzyme kinetics is that the forward reaction $v_{1}$ to form the complex is significantly larger than the reverse reaction $v_{{-1}}$ and the product-creating reaction $v_{2}$, so it is assumed that after an inner layer in which the complex builds up very quickly, ${\frac {dC}{dt}}\approx 0$. With some arithmetic, an approximate analytic solution for $C$ is found:

$C={\frac {k_{1}SE_{0}}{k_{{-1}}+k_{2}+k_{1}S}}={\frac {SE_{0}}{K_{m}+S}}$, where $K_{m}={\frac {k_{{-1}}+k_{2}}{k_{1}}}$.

The quasi-steady-state assumption also implies that substrate is converted more or less immediately into product. Thus, we have derived an expression for the velocity of the reaction:

$V={\frac {dP}{dt}}={\frac {V_{m}S}{K_{m}+S}}$, where $V_{m}=k_{2}E_{0}$, the maximum reaction rate.

A more accurate expression for the substrate and complex can be obtained by using the differential equation method of matched asymptotic expansions, which is particularly useful for singularly perturbed problems of this type, featuring an inner layer and an outer layer with changes on very different time scales. For an in-depth look at this process, the Mathematics department at UC Davis has outlined the matched asymptotic method as it is commonly performed on the Michaelis-Menten system here.