Michaelis–Menten Kinetics: How It Works

Introduction

Chemical reactions between substances differ in the rate at which they happen, or in other words, how many molecules of substrate S1 are turned into product P1 during a certain amount of time (with a unit of mol/s, for example). The basic reaction is described as follows:

S1->P1 

Since humans and other living things have thousands of reactions happening all the time, how does the body avoid overproduction and uncontrollable reactions? Why do all substrates not immediately turn into product? The short answer is enzymes. The reaction of S1 into P1, without the presence of an enzyme, might have the rate of 1 mole per 100 hours, meaning that not much is happening. Adding an enzyme provides a new reaction path with a lower energy requirement. The enzyme can easily speed up the reaction by more than a million times the original rate, making S1 turn into P1 at a rate of about 3 mol/s or more. Controlling the amount of enzyme available to the substrate thus controls the reaction speed. The enzyme E binds the substrate and creates an enzyme-substrate complex, before progressing into product.

S1+E ⇌ ES -> P1+E

This reaction chain now contains three reactions. The transformation of E and S1 into ES, the reformation from ES into E and S1, and the transformation of ES into P1 and E. Three reactions also mean three reaction rates. This is where the Michaelis–Menten kinetics comes in. This model aims to show how Michaelis–Menten kinetics can be used to represent this model in a more simplistic, yet kinetically correct way, as long as certain assumptions are met.

Explore the available scenarios by evaluating each cell in order. Evaluation is done by selecting a cell and pressing Shift+Enter.

In order to get the full experience of this example, you need the following:

Wolfram SystemModeler; a free trial download is available at www.wolfram.com/system-modeler

The model that came in the same bundle as this notebook

For more information about Wolfram SystemModeler, visit www.wolfram.com/system-modeler.

Initialize

Define the model name:

mmodel="MichaelisMenten";

Load the model:

Import[FileNameJoin[{NotebookDirectory[],"MichaelisMenten.mo"}]];

The Model

SystemModel[mmodel,"Diagram"]

In the model you now see, the upper reaction group uses pure mass action kinetics and is the model version of the following reaction:

S1+E ⇌ ES -> P1+E,

which was mentioned in the introduction. The lower reaction group is just one single reaction between S2 and P2, but uses Michaelis–Menten kinetics. While the rate of the first step:

S1+E ⇌ ES

can be described as 

v1=kOn*[S1]*[E]-kOff*[ES],

the reaction following Michaelis–Menten kinetics has a reaction rate of

(Subscript[v, MM])= (([(Subscript[V, Max])]* [S])/((Subscript[K, M])+[S]))

describing the entire reaction, i.e both the reaction described by v1 and the other reaction in the mass action model. Obviously, the Michaelis–Menten model is simpler, with fewer reactions and no intermediates, and it has an easier net formula. (Subscript[V, Max]) is the maximum rate at which substrate can be converted into product, and (Subscript[K, M]) is 

(Subscript[K, M])= (((Subscript[k, Off])+(Subscript[k, Cat]))/(Subscript[k, On])),

which is called the Michaelis constant, where (Subscript[k, On]), (Subscript[k, Cat]), (Subscript[k, Off]) are the constants for enzyme binding, product formation and substrate unbinding, respectively. 

One important prerequisite for Michaelis–Menten kinetics to work is that (Subscript[k, Cat]) is much greater than (Subscript[k, Off]) and (Subscript[k, On]). This means that the ES state, also known as a reaction intermediate, barely manages to form before moving on to becoming product and releasing the enzyme.

Simulation

Simulate the model to compare how well the Michaelis–Menten model compares to the actual reaction:

SystemModelPlot[mmodel,ImageSize->Medium]

The initial reaction rates can also be studied:

SystemModelPlot[mmodel,"Reaction Rates",ImageSize->Medium,PlotRange->{{0,2},Full}]

Using the model values for the parameters, the conditions for the Michaelis–Menten assumption are fairly well met. The two reaction kinetics correspond to each other as soon as the enzyme complex in the mass action reaction enters steady state.

One critical assumption made by Michaelis and Menten was that complex formation and disassociation occur more rapidly than product formation. By modifying the concentration of the enzyme, you can see what happens if these conditions are not met:

SystemModelPlot[
SystemModelSimulate[mmodel,{0,20},<|"ParameterValues"->{"Ec0"-> 1}|>],
ImageSize->Medium]

Analysis

To see all of this happen in real time, simulate the model and create a plot in which you can manipulate different initial conditions and see how the different substances are affected as time passes.

In order to analyze the model, a GUI for manipulating the concentrations of substrate and enzyme can be used:

plotVars={"S1.c","S2.c","P1.c","P2.c","ES.c"};
params={"kOn","kOff","kCat","Ec0"};
Manipulate[SystemModelPlot[SystemModelSimulate[mmodel,{0,20},
<|"ParameterValues"->Map[#[[1]]-> #[[2]]&,{params,{kon,koff,kcat,EZ0}}\[Transpose]]|>],plotVars,
PlotLegends->{
"Substrate 1 (MA)",
"Substrate 2 (MM)",
"Product 1 (MA)",
"Product 2 (MM)",
"Enzyme-Substrate complex (MA)"
},
Ticks->None,
PlotStyle->{
ColorData[97][1],
Directive[ColorData[97][1],Dashed],
ColorData[97][2],
Directive[ColorData[97][2],Dashed],
ColorData[97][3]}
,
ImageSize->{300,Automatic},
AxesLabel->{"Time","Concentration"}],
{{kon,2,"kOn (mol/s)"},0.01,5},
{{koff,4,"kOff (mol/s)"},0.01,5},
{{kcat,0.5,"kCat (mol/s)"},0.01,5},
{{EZ0,0.05,"Enzyme Concentration (mol)"},0.01,5},
ContinuousAction->False]

In the plot legend, mass action is abbreviated MA and Michaelis–Menten kinetics is abbreviated MM.

In the plot, you see the concentration of the substrates and products of the reaction groups, as well as the enzyme-substrate complex of the mass action model. The sliders control the magnitude of the rate constants and allow you to explore under which conditions the Michaelis–Menten model agrees with the mass action model. The states displayed are in the concentration unit of mol/l.

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