Learning Scenario – Michaelis-Menton Equation
(Vensim)
Basic Model:
Description
This is a system model of an enzyme reaction. In this reaction, each reactant
molecule must bind to an enzyme molecule in order to become a product. Enzymes
can only bind to one reactant at a time and take a fixed amount of time to convert it
to a product, so the concentration of enzymes is the limiting factor for the reaction.
The transformations of reactant and enzyme into enzyme-reactant complex, and
enzyme-reactant complex into enzyme and product, are defined by the MichaelisMenton equation. Parameters in this model determine the inputs to the equation,
which in turn determine the rate of the reaction.
Background Information
Enzyme reactions are initiated by a bond between the enzyme and the
substrate, or reactant. When the enzyme and substrate bond, the substrate is
converted into a product while the enzyme, by definition, remains untouched. The
enzyme can then bond with another molecule of substrate and continue the
reaction. Enzyme-substrate reactions are generally one-way, since the product
cannot bind to the enzyme to turn back into the reactant. However, it is possible for
an enzyme-substrate complex to devolve back into an enzyme and a substrate
without reacting. These equations are vital to biochemistry in a variety of areas,
including antibody-antigen interaction and interaction between protein molecules.
The Michaelis-Menton equation was developed by Leonor Michaelis and
Maud Menten in 1913. While studying the kinetics of the enzyme invertase, which
acts as a catalyst in the hydrolysis of sucrose into glucose and fructose, the
eponymous scientists proposed a mathematical model of the reaction under the
assumption that the enzyme concentration is much less than the reactant
concentration. The equation depends on the enzyme concentration; the “turnover
number”, which defines how quickly an enzyme can convert substrate into product;
and the half-maximum constant, which is the substrate concentration at which the
reaction rate is exactly half of maximum.
Science/Math
The fundamental principle behind this model is HAVE = HAD + CHANGE.
Each time tick in the reaction, the following things happen:
1) If there are both enzymes and substrate available, a proportion of the
enzymes and substrate will bind together to form enzyme-substrate
complexes
2) Enzyme-substrate complexes have a user-defined change of either separating
back into an enzyme and a substrate, or separating into an enzyme and a
product
3) Once a substrate has become a product, it can no longer interact with the
enzyme
Teaching Strategies
An effective way of introducing this model is to ask students to brainstorm
how an enzyme model would intuitively work. Ask the following questions:
1) If a chemical reaction requires a substrate and an enzyme to come together
in order to convert the substrate to a product, what factors would you expect
to have an effect on the reaction rate? Why?
2) Assuming that there is a relatively small amount of enzyme and a large
amount of substrate, how quickly would you expect the reaction to proceed?
Linearly, exponentially, logarithmically, etc? How do you know?
3) If the product, once created, cannot interact with the enzyme again, what
would you expect to be the long-run steady state of the reaction? Why?
Implementation:
How to use the Model
This model looks complex at first, but all of the equations are interrelated to
a high degree, making it easy to change the model with just a few inputs. The
parameters for this model are k1, km1, and k2. Their effects are as follows:
1) The parameter k1 defines the probability that an enzyme and a substrate
molecule will interact to form an enzyme-substrate complex
2) The parameter km1 defines the probability than an enzyme-substrate
complex will split back into an enzyme and a substrate
3) The parameter k2 defines the probability than an enzyme-substrate complex
will split into an enzyme and a product
All of the aforementioned parameters are manipulated before the model is
run by right clicking (control clicking on a Mac) on their labels in the diagram.
When the model is run, the parameters can also be manipulated by clicking and
dragging their respective sliders. The maximum, minimum, and step values for each
parameter are pre-set. Any changes made to the sliders take effect immediately. For
more information on Vensim, reference the Vensim tutorial at:
http://shodor.org/tutorials/VensimIntroduction/Preliminaries.
Learning Objectives:
1) Understand the relationship between enzyme concentration, substrate
concentration, and reaction speed
2) Examine the mathematics behind the reaction rate
Objective 1
To accomplish this objective, have students run the simulation and then
individually manipulate each parameter in turn. Ask the following questions:
1) From your observations, how does changing the k1 parameter affect each of
the four compounds? What happens if you make the k1 parameter extremely
large? Why does this make sense?
2) From your observations, how does changing the km1 parameter affect each
of the four compounds? Intuitively, what event does the parameter
measure? Does this have the effect you predicted?
3) From your observations, how does changing the k2 parameter affect each of
the four compounds? Does this change the shape, slope, or neither of the
enzyme graph? Why do you think this is?
Objective 2
To accomplish this objective, introduce students to the Michaelis-Menton
equation: v = kcat[E]0 * [S]/(km + [S]). In this equation, v represents the reaction rate,
kcat represents the rate at which an enzyme can convert substrate into product, E
represents the enzyme concentration, S represents the substrate concentration, and
km represents the substrate concentration at which the reaction speed is exactly half
of the maximum. Ask the following questions:
1) Are there any parallels between this equation and the factors we’ve been
working with? Which of our factors have an effect captured by this equation?
How do you know?
2) The parameter km is also commonly referred to as an inverse measure of the
substrate’s affinity for the enzyme. What does that mean? Is this parameter
captured in our model? How?
3) What factors could you change to increase the reaction rate, based on this
equation? Are these factors similar to what you predicted? Why or why not?
Extensions:
1) Model a reaction with two reactants but no enzymes required
2) Discuss what a reversible enzyme reaction might look like
Extension 1
Ask students to consider what changes they might make to the chemical
equation for this reaction in order to model two reactants rather than a reactant and
an enzyme. Ask the following questions:
1) What is the crucial difference between a reactant and an enzyme? How is
this difference modeled in our equation?
2) In a Michaelis-Menton reaction, we assume that there is a small amount of
enzyme compared to the substrate. Is this an appropriate assumption when
there are two reactants? Why or why not?
3) What would you expect to change about the Michaelis-Menton equation if we
were modeling two reactants? Why?
Extension 2
Have students think about what a reversible enzyme reaction might look like.
Emphasize that a single enzyme can only accept one substrate, so a reversible
reaction would require two separate enzymes. Ask the following questions:
1) What would you have to introduce to this model in order to make the
reaction reversible? Why?
2) In a reversible enzyme reaction, what would you expect to happen to the
concentration of reactants and products over time? What would the long-run
state depend upon?
3) Does the Michaelis-Menton equation apply to a reversible reaction? If not,
what changes would you make to the equation to make it apply?
Related Models
Generic Chemical Dynamics
http://www.shodor.org/talks/ncsi/vensim/index.html
This is a simpler reaction model with just a single two-stage chemical
reaction. Unlike in the Michaelis-Menton model, there is no enzyme and just two
reactants. The reaction can either go to completion or go to a long-run equilibrium,
depending on the reversibility of the reaction. This model is a great resource to
check students’ work converting the Michaelis-Menton model to one without an
enzyme. Running this model can also reveal the different ways in which reactions
can be modeled, and the different conclusions that result.
Rabbits and Wolves
http://www.shodor.org/interactivate/activities/rabbitsandwolves
This model, on the surface, has very little to do with chemical reactions.
Rabbits and wolves roam a field of grassland where the rabbits eat the grass and the
wolves eat the rabbits. Both species then reproduce according to specific rules.
However, there are certain parallels that can be drawn between these models if the
rules are set up correctly. If you disallow reproduction and death from old age,
rabbits and wolves can be thought of as a model of an enzyme reaction. The wolves
are the enzymes, the rabbits are the substrate, and the result of their meeting is the
product, a dead rabbit. Looking at the time series tracking the number of rabbits
over time, students can compare rates of change in this situation to those in an
enzyme reaction. With repeated trials, it is even possible to estimate the MichaelisMenton constants that would yield such a graph.