Problem Statement

This module on cobra venom and its behavior in the human body was originally given as an open-ended problem, or OEP, during Winter Semester 1994 at the University of Michigan. In the spirit of truely open-ended problems, students were simply asked to investigate the effects of being bitten by a poisonous snake. A base case for the problem and some possible starting suggestions were given to the students, but the emphasis was on creativity in exploring the problem.

 

Exploring the Base Case, Part 1 of 4

Initially, you should investigate the case where a human is bitten by a poisonous snake, but no antivenom is injected. Plot the fraction of free sites in the body as a function of time. You should be able to verify the time it would take for 1/3 of the receptor sites to be blocked by venom, which would then result in the death of the victim by respiratory failure.

Look at the changes that occur when antivenom is injected: How long can you wait to inject the antivenom? How much antivenom should be injected into the victim? Explain the behavior of your graphs.

First we'll do what the problem statement suggested, and we'll try to prove the claim that 1/3 of the fraction of free sites will be covered in 1/2 hour. (Remember that no antivenom will be injected to save this person's life. Cruel, aren't we?) To do this, we'll want to make use of Polymath.

Additional Information:

A cobra typically injects 2.0 x 10-7 moles of venom per bite. Based on the fact that the average human has 40 dm3 of body fluid, the initial concentration of venom in the blood stream is 5.0 x 10-9 M.

Rate Constants:
Constants1 Constants2
Initial Conditions:
at time = 0, the fraction of receptor sites = 1

Base Case Equations

Here are the equations we'll enter, based on the additional information above and the reaction information we were given earlier:

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NOTE: We are making two major assumptions about this reaction system (i.e., the venom and antivenom interacting in a human body):

  1. it can be modeled as a batch reactor, and
  2. the reactor is well-mixed.

These may or may not be appropriate assumptions to make. That is for you to determine in your exploration of this problem.

As it turns out, we are left with one term, CS0, which we must solve for first. CS0 is the initial concentration of free sites, and to find it, we must solve backwards from our solution, which demands that 1/3 of the free sites be occupied by venom after 1/2 hour. fS is the fraction of free sites available, so it should be equal to 2/3, or 0.667, after 1/2 hour. With a little trial and error (guessing the value of CS0 and solving for fS until it equals 0.667), we can determine that CS0 must equal 5 x 10-9 M, which just so happens to be equal to the concentration of venom in the blood stream, CV, at time zero.