COMPUTATIONAL MODELING AND SIMULATION AS ENABLERS FOR BIOLOGICAL DISCOVERY 129
recognizing that what is “essential” cannot be determined once and for all, but rather depends on the
class of questions under consideration.
5.3.2 Hybrid Models,
Hybrid models are models composed of objects with different mathematical representations. These
allow a model builder the flexibility to mix modeling paradigms to describe different portions of a
complex system. For example, in a hybrid model, a signal transduction pathway might be described by
a set of differential equations, and this pathway could be linked to a graphical model of the genetic
regulatory network that it influences. An advantage of hybrid models is that model components can
evolve from high-level abstract descriptions to low-level detailed descriptions as the components are
better characterized and understood.
An example of hybrid model use is offered by McAdams and Shapiro,^29 who point out that genetic
networks involving large numbers of genes (more than tens) are difficult to analyze. Noting the “many
parallels in the function of these biochemically based genetic circuits and electrical circuits,” they
propose “a hybrid modeling approach that integrates conventional biochemical kinetic modeling within
the framework of a circuit simulation. The circuit diagram of the bacteriophage lambda lysislysogeny
decision circuit represents connectivity in signal paths of the biochemical components. A key feature of
the lambda genetic circuit is that operons function as active integrated logic components and introduce
signal time delays essential for the in vivo behavior of phage lambda.”
There are good numerical methods for simulating systems that are formulated in terms of ordinary
differential equations or algebraic equations, although good methods for analysis of such models are
still lacking. Other systems, such as those that mix continuous with discrete time or Markov processes
with partial differential equations, are sometimes hard to solve even by numerical methods. Further, a
particular model object may change mathematical representation during the course of the analysis. For
example, at the beginning of a biosynthetic process there may be very small amounts of product so its
Box 5.3
An Illustration of “Essential”Consider the following modeling task. The phenomenon of interest is a monkey learning to fetch a banana
from behind a transparent conductive screen. The first time, the monkey sees the banana, goes straight ahead,
bumps into the screen, and then goes around the screen to the banana. The second time, the monkey, having
discovered the existence of the screen that blocks his way, goes directly around the screen to the banana.To model this phenomenon, a system is constructed, consisting of a charged ball and a metal sheet. The
charged metal ball is hung from a string above the banana and then held at an angle so the screen separates
the ball and the banana. The first time the ball is released, the ball swings toward the screen, and then touches
it, transferring part of its charge to the screen. The similar charges on the screen and the ball now repel each
other, and the ball swings around the screen. The second time the ball is released, the ball sees a similarly
charged screen and goes around the screen directly.This model reproduces the behavior of the monkey in the first instance. However, no one would claim that it
is an accurate model of the learning that takes place in the monkey’s brain, even though the model replicates
the most salient feature of the monkey’s learning consistently: both the ball and the monkey dodge the screen
on the second attempt. In other words, even though it demonstrates the same behavior, the model does not
represent the essential features of the biological system in question.(^29) See H.H. McAdams and L. Shapiro, “Circuit Simulation of Genetic Networks,” Science 269(5224):650-656, 1994.