Calculus and Pancreatic Cancer Homework Exercises

Questions adapted from: Dr. Stephan Gysin of UCSF (guest speaker)

A chemical reaction proceeds by combining two reactants, A and B, and getting the product C as follows:

Although the rate of reaction is determined experimentally, it often follows a model in terms of reactants or product. The rate at which a given chemical reaction proceeds can either be expressed either by the rate of disappearance of reactants

or the rate of formation

where [A],[B], and [C] are concentrations that can change over time.

If a product is being formed and depleted at the same time the overall rate is the rate of depletion subtracted from the rate of formation

  1. A typical enzyme catalyzed reaction is given by the following chemical equation:

E is the free enzyme, S is the substrate, ES is the enzyme in complex with the substrate and P is the product.

The rate of formation of ES is given by:

d[ES]f/dt = k1[E][S]

whereas the rate of breakdown of ES is given by:

d[ES]b/dt = (k-1 + k2)[ES]

What is the overall rate of reaction for [ES]?

  1. It is often assumed that the chemical reaction above has a steady state kinetics, i.e. [ES] = constant. What does this say about the equation you found in #1?
  1. [Etot] is the total enzyme concentration and remains constant. The enzyme is either bound to the substrate or is free and can be expressed: [Etot] = [E] + [ES]

Use this information along with what you found in #2 to find [ES] in terms of [S].

  1. The rate of product (P) formation is given by: d[P]/dt = k2[ES]

Km is the Michaelis constant which is given by:Km = (k-1 + k2)/k1

The maximal rate of product formation d[P]/dt = Vmax is attained when the enzyme sites are saturated with substrate, i. e. [S] > Km. Under these conditions Vmax = k2[Etot].

Derive an equation of d[P]/dt = v as a function of [S] including Vmax and Km using the assumptions given above. The result will be the Michaelis-Menten equation.

  1. Use the graphing techniques of Chapter 5 in your text to sketch the graph of v vs. [S] over the domain (-∞,∞).
  1. Look at an actual Michaelis-Menton graph, is it different from what you obtained in #5? You can look at the presentation for this graph at: or by searching on-line for “Michaelis-Menton”. Give a practical reason for why this might be.
  1. What happens to the reaction rate equation at high substrate concentrations?
  1. Assume that an enzyme-catalysed reaction follows Michaelis-Menten kinetics with a Km of 1 M. The initial velocity is 0.1 M/min at 10 mM substrate. Calculate the initial velocity at 1 mM, 10 M and 1 M substrate. If the substrate concentration were increased to 20 mM would the initial velocity double? Why or why not?
  1. In about 30-50 words, please describe how and whether the field trip has helped your learning. Please submit the answer to this question on an individual basis and not as a group.