higher-order mass-action, enzyme kinetic equations, or simply
Boolean logics, depending on the knowledge gained for an individ-
ual reaction.
In our research, we have used the linear mass-action approach
and the law of conservation to model the dynamical responses of
innate immune [11–14] and cancer response [15–17] to distinct
perturbations (closed system approximation). In these signaling
studies, the respective models have revealed missing molecular
species, novel bypass pathways, and crosstalk mechanisms that
were experimentally verified. In addition, for TNF and TRAIL
signaling, the models have aided in identifying crucial molecules
to regulate proinflammatory and apoptotic responses, respectively.
From the successes of these studies and that of many others, it is
evident that highly complex biological networks, upon external
perturbation, can stably process their downstream information
through linear response waves [4, 18, 19].
It is interesting to note that numerous closed system linear
models have generated insightful results, especially when dealing
with population-averaged metabolic or signaling networks, despite
the fact that living systems are constantly exchanging matter and
energy to the surroundings. As such, organisms or cells should be
considered to exist far from equilibrium to achieve biological order
[20]. One important example is the ability of bacteria to exchange
pheromones during environmental threats, such as antibiotic treat-
ments, to form biofilms that are highly organized structures resis-
tant to the therapeutic intervention [21]. The biofilm example
demonstrates that the cooperative behavior of organisms can be
very different to their individual response. Thus, using the ergodic
principle or predictive deterministic approaches to understand the
majority of cellular behaviors can be questionable, and this issue has
been debated from time to time.
The following sections are devoted to more complex response
that do not follow the closed system approximation and require
more sophisticated, yet simple, nonlinear differential equations to
understand their dynamical response.3 Nonlinear Dynamics
3.1 Periodic
Oscillations
The mass-action and enzyme kinetics equations have been largely
used to study stable equilibrium conditions, where steady-state
levels (i.e., dXdti¼0 ) or clear formation and depletion (linear)
response waves of molecular species are observed. Under several
other conditions, self-organizing oscillatory behaviors and multi-
stable levels have been realized. Under these conditions, nonlinear
differential equation approaches have been investigated.
Periodic behaviors or biological rhythms, such as sleep and
menses cycles, have been evident since the evolution of life.Complex Biological Responses Using Simple Models 181