aerospace industries. Similarly, in thermodynamics related to gas
flows, the ideal gas law (the popularPV¼nRTequation), although
poses several limitations, is still widely used in the chemical indus-
tries as a good approximation to study the behavior of many gases
under various conditions. The lessons from other disciplines over
the last two centuries have indicated that simple theoretical founda-
tions under a “window” of operating conditions can be highly
beneficial to model realistic behaviors [3, 4].
In biology, the goal to simplify complex biochemical networks
using a reductionist view is also not new. In the early part of the last
century, Victor Henri, Leonor Michaelis, and Maud Menten thor-
oughly investigated enzymatic biochemical reactions, in vitro, and
developed the hyperbolic rate equation that we now popularly call
the Michaelis-Menten enzyme kinetics. Using this method, func-
tional networks of the energy metabolism, such as the glycolysis
and the Krebs cycle, have been widely studied. However, the earlier
models were plagued with the dilemma where increased accuracy in
model prediction required detailed knowledge of in vivo reaction
parameters that were too difficult to measure precisely. Most, if not
all, studies adopted in vitro experiments to determine the parame-
ter values of reaction species from an artificial environment where
the species were deliberately purified from its physiologic neigh-
bors. There have been various reports that claim the kinetic para-
meters determined through in vitro and in vivo experiments can
differ by several orders of magnitudes [5]. As a result, when com-
bining these errors into the model, the final predictions could differ
by several orders of magnitude. For example, the steady-state con-
centration of the glycolytic metabolite 3-phosphoglycerate inTry-
panosoma bruceiwas under-predicted by an order of 7 [6].
Despite the difficulty posed by adaptive living systems, numer-
ous mathematical techniques have been developed and used to
decipher major system-level properties in development, differenti-
ation, growth, aging, and the immune response. In this chapter,
fundamentals of simple linear models for understanding cellular
response that follow formation and depletion waves are introduced
(Subheading2). This is followed by a review on nonlinear models
developed to recapture oscillatory responses observed in cells (Sub-
heading3). Linear stability analysis that could be used to study the
robustness of multi-stable states of living systems is also introduced
(Subheading4). A brief look at the reaction-diffusion equations for
spatio-temporal pattern analysis is presented in the penultimate
section (Subheading5), followed by comments on stochasticity
and heterogeneity more recently observed in single-cell systems
(Subheading6).
Complex Biological Responses Using Simple Models 173