Calculus: Project 49

Project 35 discusses linear drug models that accurately model many drugs. Beer, cocaine, and vitamin C are exceptions. Ethanol and cocaine are metabolized by an enzyme reaction and absorption of vitamin C is nonlinear. All three are well modeled by a limiting case of the general enzyme equations known as the Michaelis-Menten equation.

49.1 Enzyme-Mediated Reactions "Enzymes" are chemicals that bind to a "substrate" and allow an irreversible reaction to take place creating a "product." The reaction frees the enzyme to do its job again but does not use it up. The substrate will not react, or only react very slowly, unless it is bound by the enzyme. Schematically, our enzyme reaction looks as follows:

where S represents substrate, Z represents enzyme, C represents the substrate-enzyme complex, and P represents the product.

49.2 Molar Concentration and Reaction Rates

We want to write equations that describe how these reactions proceed. The amounts of the various substances change in time, so time derivatives of a measure of each will give us such a description. In this case a good choice of units to measure the substances will greatly simplify the description. A natural choice might be to measure mass, say in grams, or the mass concentration of each substance, since a typical reaction takes place in liquid. This is not the best choice. Units of molar concentration make our reaction balances easiest to describe with derivatives.

A mole of a compound is Avagadro's number, , of molecules of the substance. This is the atomic weight of the compound in grams, so, for example, a mole of carbon 12 weighs 12 grams. We will use units of "molars" or moles per liter in our examples, specifically millimolars. The reason this choice of units is best is that one molecule of enzyme combines with one molecule of substrate to form one molecule of complex.

VARIABLES

The choice of molar concentrations means that the rates on the left side of the schematic reaction satisfy

and on the right side

Enzyme is "lost" or bound on the left and "gained" or released on the right.

PARAMETERS

Each of the rate constants is proportional to the product of the concentration of the reactants, so

As a differential equation the enzyme-substrate forms and unforms by

The reaction of complex to product is given by

Three things change the concentration of complex. It can disassociate back into substrate and enzyme, it can react into product and enzyme, or substrate and enzyme can combine to form it:

Since enzyme is either free and part of z or bound and part of c,

1. Prove that the sum a=s+c+p is constant. Hint: The derivative of a constant is zero.
2. Prove that b=z+c is constant.
3. Use these relationship to prove that the four equations for , , , and reduce to just the equations

Explain how you would find z[t] and s[t] if you were given a solution to these two equations.

4. How are a and b determined at the beginning of a reaction? Write your answer as an augmented initial value problem,

49.3 The Briggs-Haldane Dynamics Approximation
In some reactions, after a short period, the NET rate of formation of complex is small compared with the flow of substrate to complex and complex to product, so .
The Briggs-Haldane Initial Value Problem
- 4. How can we find p[t] and z[t] from the solution of the Briggs-Haldane IVP?
  Problem 49.1 Briggs-Haldane Numerical Experiments
  Compare the dynamics of the solution of the full enzyme dynamics in Hint 49.2.1 with the Briggs-Haldane initial value problem in the case where h=?, k=?, r=? and in the case h=??, k=??, r=??
  
  49.4 The Michaelis-Menten Dynamics Approximation
  Usually, we use a great deal of substrate and only a little enzyme to perform a reaction. The Michaelis-Menten approximation arises when the amount of substrate is large compared to the amount of enzyme and the product reaction is slow compared with the disassociation, r<<h. In this case, s>>z, c would rather go back to s and z, so c and p are small. Thus, the constant . We always have b=z+c, constant, so z=b-c.
  If k and h are large compared with r, the complex used to produce product will be quickly replaced and c will stay nearly constant. While the net concentration of c is approximately in equilibrium,
- Michaelis-Menten Initial Value Problem
  - Comparison of Briggs-Haldane and Michaelis-Menten Dynamics
    How do the Briggs-Haldane and Michaelis-Menten approximations compare? For example, under the Michaelis-Menten assumptions, how do the constants and compare? Under the Michaelis-Menten assumptions, can the Briggs-Haldane assumptions also hold? What about the converse?
    Mathematically, you can explore the differences:
    Problem 49.2 Michaelis-Menten Numerical Experiments
    Compare the dynamics of the solution of the full enzyme dynamics in Hint 49.2.1 and the Briggs-Haldane approximation with the Michaelis-Menten initial value problem in the case where h=?, k=?, r=? and in the case h=??, k=??, r=??
    
    49.5 Blood Ethanol
    Blood alcohol (ethanol) at high concentrations like 100 millimolars causes vomiting and inability to walk. Alcohol is metabolized by an enzyme called "alcohol dehydrogenase." The constant K for this enzyme is about 3 millimolars.
    
    Figure 49.1: Sobering up for 4 Hours
  - Hockey Sticks
    - 3. Why do the pharmacokinetics books call these curves "hockey stick curves"?
      Problem 49.3
      Look up the "linear" sobering rate for blood ethanol and adjust the constants above appropriately.
      
      49.6 Blood CO₂
      Carbon dioxide (CO₂) is closely related to blood pH (acidity). There is an approximately constant concentration of CO₂ in the blood of about 1.35 millimolars. One mechanism that keeps this level constant is a reaction with an enzyme caled "carbonic anhydrase." The constant K for this enzyme is about 12 millimolars. The average concentration of the enzyme is about 0.001 and the rate constant is on the order of 10⁶ per second, one of the largest enzyme rate constants.
    - Exponential Approximation
      - 3. Compare the exponential solution with the solution to the Michaelis-Menten initial value problem when K=12 and rb=1000.
        
        Figure 49.2: CO₂ Concentration in the Blood
        Problem 49.4 A Visit to the Chemistry Library
        Find more exact values of the constants for the carbonic anhydrase reaction.
        Find an intermediate speed reaction where Michaelis-Menten dynamics lie between hockey sticks and exponentials.
        
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