BMEN E3500, fall 1998

Lecture Notes

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Lecture Number 4

Date: 9/27/98

  • 1) Chemical Reactions in Compartments
  • a) Basic ideas
  • i) It is important to remember that a reaction does not create or destroy mass. The reaction transforms the chemical identity of a given mass. There must be at least one reactant and at least one product and the mass (but not necessarily the molar) sum of the reactants used must equal that of the products formed.

    ii) Thus the basic compartment equation for "A" must be accompanied by at least one other equation: if "A" is the reactant, then there is a product. If "A" is a product, there must be a reactant.

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    b) There are three ways of avoiding the consequences of having at least one companion to the "A" equation:

  • i) "A" is a product and the reactant from which A is made exists in great excess. Then the source of "A" is depleted, but the depletion is not noticeable.

    ii) "A" is a reactant that is depleted and it is only the depletion that is important in the problem. Product is formed but its presence is not of concern. This simplification also requires that the product does not undergo a reverse reaction that could form a significant amount of "A".

    iii) "A" is affected by reaction, but the reaction is fast and reversible so that the relationship between A and the other material is governed not by a rate expression but by equilibrium.

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    c) If these conditions are not met, a reaction-coupled set of equations results.

  • i) Differential equation if conditions are transient.
  • (1) Alternative formulation: separate set of compartments for each component. Reaction becomes an intercompartment exchange.
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    ii) Algebraic equations if steady state.

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    d) Flow of energy and chemical transformation in biological systems is regulated by enzymes.

  • i) Enzymes are true catalysts. They do not affect a reaction equilibrium, only the rate at which (in a closed system) equilibrium is approached.

    ii) The escalator model of enzyme kinetics.

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  • iii) Enzyme kinetic formulations.
  • (1) The elementary mechanism for irreversible (i.e. equilibrium lies far to the right) enzyme conversion of substrate S involves two steps:

    (2) Following Bailey and Ollis (Bailey, J. and Ollis DF, "Biochemical Engineering Fundamentals", 2nd ed. McGraw-Hill, 1986, Section 3.2.1) we designate the concentration of substrate as s, enzyme as e, complex as (es) and product as p and the reaction velocity as v. The dissociation constant and reaction velocity are defined:

    • The last expression records the fact that the sum of the free (e) and bound (es) enzyme present at any time must equal the amount of enzyme originally provided e0. Note that the reaction velocity is one of the factors contributing to dp/dt. Contributions from convection and diffusion are also possible.

    (3) Solving the first expression for (es) in terms of s and e and then substituting (e0 - (es)) for e, we have an expression for (es) in terms of the total enzyme concentration e0 and s.

    • This expression is linear for values of s that are small (compared to the value of Km, which has the units of concentration. The expression becomes asymptotically equal to the constant vmax when s is large. Note that the dependence of rate on enzyme concentration is always simply proportional while the S dependence varies from linear (proportional) to invariant as the concentration of S increases. In terms of the escalator model discussed above, large values of s conform to a "loaded" escalator. Delivering more people to the bottom of the escalator will not cause it to deliver many more people to the next floor. Raising the value of s much above the value of Km will not much increase the amount of product formed.

    If the enzyme reaction is reversible, the second reaction (in paragraph 1, above) is reversible. The equation set becomes:

    This small change gives a considerably different result (which the student is responsible for deriving):

    while the value of vs is vmax and the value of Ks is Km.

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