10 Equations in Biology:Michaelis-Menten Kinetics

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 ESkf

kr

E + Pkcat

Variables[E]: free enzyme molecules[S]: free substrate molecules[ES]: enzyme-substrate complexes[P]: free product molecules

Parameterskf , 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 proto-mathematical understanding of a system before introducing formal math.

$ in checking account on Sept. 1st

=deposits made in August

–

Example: Balancing a checkbook

$ in checking account on

Aug. 1st

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

# of free S molecules binding to free E = kf [E]t [S]t

# of ES complexes splitting into E + S = kr [ES]t

conc. of enzyme-substrate complexes

conc. of free enzyme conc. of free substrate

by Law of Mass Action

Formal Equations for Enzyme Kinetics Model

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 influencethe reaction’s velocity? Why?

Further ExplorationEnzyme 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 Michaelis-Menten curves of enzymes w/low, medium, and high KM.

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 scrollbarsto adjust estimatesof vmax and Km.

Estimating Reaction Constants:Linear Methods

Goal: Convert the Michaelis-Menten equation into a form that linearizes the best-fit curve.

ExampleLineweaver-Burk plot:

1/v vs. 1/[S]0

Formula = …?

Should be able to write similar formulas for other plots (H-W, E-H)

Estimating Reaction Constants:Evaluating Diff. Methods

On what basis might one methodbe 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 understandingof a system, and to ground abstract discussions

• Lesson structure: what aspects of this seminar did you find most/least crucial?