Michaelis-Menton Learning Scenario (Web)

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 Michaelis-Menton 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:

Every 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
3. 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
4. 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

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

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 Michaelis-Menton constants that would yield such a graph.