Biomedical Engineering Reference
In-Depth Information
This chapter focuses on chemical reactions that occur inside and outside a cell following
the law of mass action—that is, the rate of accumulation is proportional to the product of
the reactants. These chemical reactions support all functions needed to support life and
involve such activities as the synthesis of hormones and proteins, muscle contraction, respi-
ration, reproduction, neural signaling, and many other reactions. While the law of mass
action is useful, it is not appropriate for all chemical reactions. In some cases, the exact
chemical reaction mechanism is not known, and the law of mass action doesn't work.
The law of mass action creates models that are nonlinear and possibly time-varying. In
addition, the rate of accumulation is profoundly impacted by temperature, such that an
increase in temperature increases the reaction rate. The reactants are assumed to be uni-
formly distributed in the compartment, the probability of a collision depends on the con-
centration of the reactants, and such collisions are sufficient to create the products. Since
the models here are nonlinear, we typically use SIMULINK to solve these problems. How-
ever, we will present analytical solutions to some special nonlinear cases in this chapter.
Catalysts are substances that dramatically change reaction rates. These substances are
generally present in small amounts and are not consumed in the reaction. Their function
is to decrease the amount of time to reach steady state in a chemical reaction. Consider
two reactants that spontaneously create a product at room temperature but do so at a very
low rate. In the presence of an appropriate catalyst, the speed of the reaction is dramatically
increased, and the time to reach steady state is decreased. Enzymes are protein catalysts
used in biological reactions that regulate and control most processes in the body. Enzymes
increase the reaction rate by thousands or even trillions and are reactant specific. Typically,
the enzyme concentration is rather small in relation to the reactant. While the reaction rate
is profoundly increased in the presence of an enzyme, the overall energy used to form the
product does not change.
In this chapter, we first examine simple chemical reactions and then enzyme kinetics.
Next, we look at quasi-steady-state approximations, the Michaelis-Menten elimination,
and enzyme regulation. We finish by examining important processes, such as the transport
of oxygen and carbon dioxide through the circulatory system to the cell, the Na-K pump,
and cellular respiration.
8.1 CHEMICAL REACTIONS
Consider the following single-stage chemical reaction
K
A
þ
B
!
ð
8
:
1
Þ
P
in which chemicals
K.
Equation (8.1) is known as a stoichiometric equation that lists the number of reactants on
the left side necessary to form the product on the right side. Conservation of mass requires
that the total quantity of reactant
A
and
B
react to form the product
P
, with reaction rate constant
in the reactant and the
product, and likewise, the same is true with reactant B. The stoichiometric equation does
not describe the dynamics or kinetics of the chemical reaction—that is, the time course of
the reaction that may be very fast or very slow. The kinetics of the reaction is written
A
must equal the quantity of
A
