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.

Cases

Initially, we should investigate the case where a human is bitten by a poisonous snake, but no antivenom is injected. We can plot the fraction of free sites in the body as a function of time to verify the time it would take for 85 to 90% of the receptor sites to be blocked by venom (which would then result in the death of the victim by respiratory failure).

We can also 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?

To organize these thoughts, we will divide our problem into these cases:

1. Case 1: Worst Case Scenario
   You are lost in a jungle and suddenly get bit by a King Cobra after carelessly treading over a pile of old leaves. You are hours away from any civilization and don't have any antivenom on hand. In this case, we will try to prove the claim that 85 to 90% of the fraction of free sites will be blocked in half an hour. Remember that no antivenom will be injected to save your life. Cruel, aren't we? We will also explore the effects of venom on the twitch height. Once the twitch height is lower than a certain threshold, the human respiration system will cease to function normally.


2. Case 2: Wow, you were ready for that weren't you?
   You were so ready for that cobra bite that you injected antivenom right when the cobra bit you! In this case, we will explore the effects of immediately injecting antivenom into the bloodstream after a cobra bite. We will also analyze the effects of an immediate injection of antivenom on the twitch height.


3. Case 3: Oh no! Where's the closest stock of antivenom?
   You decided to explore some ancient ruins in the rainforests of Thailand and you accidentally step on and get bit by a cobra. You are a few minutes away from your car, where the antivenom and first aid kit is located. In this case, we will explore the effects of injecting antivenom 28 minutes after a cobra bite which is the point of no return as explained later. We will also explore the effects of delayed injections of antivenom on twitch height.


4. Case 4: Oops, that was a little too much antivenom
   Learning from your last encounter, you now have antivenom prepared in case a cobra were to bite you. After accidentally slipping while climbing a hill, you feel something strike your left leg. You haphazardly reach into your pocket and inject antivenom into your thigh. Right after the antivenom had been administered, you notice a small young cobra slither away. It turns out that had just accidentally injected more antivenom than necessary. In this case, we will explore what happens when too much antivenom has been injected.


5. Case 5: Well that escalated quickly
   On another trip to the jungle, you accidentally lost footing and fall back onto your backpack when you feel a painful prick on your back. You get up and notice that the antivenom had pierced through the backpack and was inadvertently administered into your bloodstream. In this case, we will explore the effects of an accidental injection of antivenom.


6. Case 6: Your Turn.
   In this case, it is your turn to explore whatever you've wondered about while going through this problem. In the spirit of open ended problems, question the methods used in this module, determine whether the reaction mechanisms make sense, invent a needle of some sort to inject the antivenom, anything you want!

Assumptions

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.

Determining C_S0

As it turns out, we are left with one term, C_S0, which we must solve for before we are able to analyze this problem. C_S0 is the initial concentration of free sites, and to find it, we must solve backwards from our solution, which demands that 85 to 90% of the free sites be occupied by venom after 30 minutes.

f_S is the fraction of free sites available, so it should be equal to 20 to 25% after 30 minutes. With a little trial and error (guessing the value of C_S0 and solving for f_S until it equals ~0.80), we can determine that C_S0 must equal 5 x 10^-9 M, which just so happens to be equal to the concentration of venom in the blood stream, C_V, at time zero.

The Polymath code we used to determine C_S0 is avaliable here:
Download the Code (.pol)

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. (Recall that 1 M = 1 mol/dm³)

Rate Constants:

k_v = 3*10⁹ dm³/(mol*hr)

k_A = 2*10⁸ dm³/(mol*hr)

k_iA = 1 hr^-1

k_ov = 0 hr^-1

k_op = 0.3 hr^-1

k_sv = 6*10⁸ dm³/(mol*hr)

k_sA = 6*10⁸ dm³/(mol*hr)

k_p = 1.2*10⁹ dm³/(mol*hr)

k_oA = 0.3 hr^-1

Initial Conditions:

1. at time = 0, the fraction of free receptor sites = 1
2. the fraction of sites occupied by venom = 0
3. the fraction of sites occupied by antivenom = 0
4. concentration of venom in the blood = 5.0 x 10^-9 M
5. concentration of antivenom in the blood = 0 M
6. concentration of venom-antivenom product in the blood = 0 M