10 Equations in Biology:
Michaelis-Menten Kinetics
[E]t+1 = [E]t - kf [E]t [S]t + kr [ES]t + kcat [ES]t
[S]t+1 = [S]t - kf [E]t [S]t + kr [ES]t
[ES]t+1 = [ES]t + kf [E]t [S]t - kr [ES]t - kcat [ES]t
[P]t+1 = [P]t + kcat [ES]t
Outline
1. Brief intro to enzyme kinetics
2. From intuitive conceptions to a formal model
3. Building interactive models in Excel
4. From a mathematical model to biological insights
Intro to Enzyme Kinetics
E+S
kf
kr
ES
Variables
[E]: free enzyme molecules
[S]: free substrate molecules
[ES]: enzyme-substrate complexes
[P]: free product molecules
kcat
E+P
Parameters
kf , kr , kcat : reaction rates
Writing “Word Equations”
• A word equation is a qualitative description
of the system’s major processes
• Goal: help students express their protomathematical understanding of a system
before introducing formal math.
Example: Balancing a checkbook
$ in checking
account on =
Sept. 1st
$ in checking
account on
Aug. 1st
?
+
deposits
made in
August
–
withdrawals
made in
August
Word Equations for
Enzyme Kinetics Model
# of free S
molecules
at time t +1
=
# of free S
molecules
at time t
–
# of free S
molecules
binding to
free E
+
# of ES
complexes
splitting
into E + S
Write similar word equations for the other 3 variables.
Translate Word
Equations into
Formal Equations
# of free S molecules
at time t
= [S]t
conc. of enzymesubstrate complexes
# of ES complexes
splitting into E + S
= kr [ES]t
conc. of free enzyme
 conc. of free substrate
# of free S molecules
binding to free E
= kf [E]t [S]t
by Law of Mass Action
Formal Equations for
Enzyme Kinetics Model
[E]t+1 = [E]t - kf [E]t [S]t + kr [ES]t + kcat [ES]t
[S]t+1 = [S]t - kf [E]t [S]t + kr [ES]t
[ES]t+1 = [ES]t + kf [E]t [S]t - kr [ES]t - kcat [ES]t
[P]t+1 = [P]t + kcat [ES]t
Modeling in Excel
Why Excel rather than more powerful tools?
• easily available on most campuses
• widely used = transferable proficiency
• nonthreatening to students
• makes students work directly with the equations:
no dodges via the user interface!
Implementing a
Model in Excel
✓ 1. Write a series of recursion equations.
2. Set up Excel sheet with time in Column A,
other variables in next columns.
3. Enter and name model parameters;
enter variables’ starting values.
4. Enter recursion equations for each variable.
5. Copy and paste for subsequent time steps.
6. Graph results appropriately.
Structured Exploration:
Understand the Model
1. How would you measure the reaction’s velocity?
How does velocity change over the course of the
reaction, and why?
2. How does each rate constant influence
the reaction’s velocity? Why?
Further Exploration
Enzyme Kinetics – Multiple Runs.xlsx
• What happens to velocity
as [S]0 increases? Why?
• Estimate vmax
for this enzyme.
Further Exploration
• Biological interpretation of KM?
• Characterize the MichaelisMenten curves of enzymes
w/low, medium, and high KM.
KM º (kr + kcat ) / kf
Estimating
Reaction
Constants
• What is the shape of
the M-M curve?
• Nonlinear regression:
Fit the appropriate
curve to the M-M plot
by minimizing the sum
of squared residuals.
• Use built-in scrollbars
to adjust estimates
of vmax and Km.
Estimating Reaction Constants:
Linear Methods
Goal: Convert the Michaelis-Menten
equation into a form that
linearizes the best-fit curve.
Example
Lineweaver-Burk plot:
1/v vs. 1/[S]0
Formula = …?
Should be able to write
similar formulas for
other plots (H-W, E-H)
v max [S]
v=
KM + [S]
Estimating Reaction Constants:
Evaluating Diff. Methods
On what basis might one method
be considered “better” than others?
• Each method’s
sensitivity to
estimation error?
• Each method’s
ability to detect
such error?
Beyond Simple Enzyme Kinetics
How could you modify the model to describe…
• Influx or efflux of reaction components?
• Reversible, competing, or multi-step reactions?
• Totally different biological systems
(pop. dynamics, epidemiology, pharmacokinetics, …)?
Preview: Agent-Based Models
• Simulate &
visualize behavior
of each indiv.
molecule
• Focus of next
week’s seminar
(Nernst equation)
Take-home message
• Build equations to clarify each term’s
meaning and identify key assumptions
• Explore models to test understanding
of a system, and to ground abstract
discussions
• Lesson structure: what aspects of this
seminar did you find most/least crucial?