Catalyzing Inquiry at the Interface of Computing and Biology

COMPUTATIONAL MODELING AND SIMULATION AS ENABLERS FOR BIOLOGICAL DISCOVERY 155

(about 200). Such a result may be numerological coincidence or rooted in the fact that nearly all cells in
a given organism (even across eukaryotes) share the same basic housekeeping mechanisms (metabo-
lism, cell-cycle control, cytoskeletal construction and deconstruction, and so on), or it may reflect phe-
notypic structure driven by the large-scale connectivity in the overall genetic regulatory network. More
work will be needed to investigate these possibilities.^76 Box 5.10 provides one view on experimental
work that might be relevant.

To illustrate the potential value of Boolean networks as a model for genetic regulatory net-
works, consider their application to understanding the etiology of cancer.^77 Specifically, cancer is

Box 5.9

Finite-state Automata and a Comparison of Genetic Networks and Boolean Networks

In Kaufmann’s Boolean representation of a genetic regulatory network, there are N genes, each with two states

of activity (expressed or inhibited), and hence 2N possible states (i.e., sets of activities) in the network. The

number of possible connections is combinatorial in N and K. Starting at time t, each gene makes a transition

to a new state at time t + 1 in accord with the transition rule and the K inputs that it receives. Thus, the state

of the network at a time t + 1 is uniquely determined from its state at time t. The trajectory of the network as

t changes (i.e., the sequence of states that the network assumes) is analogous to the process by which genes

are expressed.

This network is an instantiation of a finite-state automaton. Since there are a finite number of states (2N), the

system must eventually find itself in a state previously encountered. Since the system is deterministic, the

network then cycles repeatedly through a fixed cycle, called an attractor. Every possible system state either

leads to some attractor or is part of an attractor.

Different initial conditions may or may not lead to different attractors. All of the initial conditions that lead to

the same attractor constitute what is known as a “basin” for that attractor. Any state within a basin can be

exchanged with any other state in the same basin without changing the behavior of the network in the long

run. In addition, given a set of attractors, no attractor can intersect with another (i.e., pass through even one

state that is contained in another attractor). Thus, attractors are intrinsically stable and are analogous to the

genetic expression pattern in a mature cell.

An attractor may be static or dynamic. A static attractor involves a cycle length of one (i.e., the automaton

never changes state). A dynamic attractor has a cycle length greater than one (i.e., a sequence of states repeats

after some finite number of time increments). Attractors that have extremely long cycle lengths are regarded as

chaotic (i.e., they do not repeat in any amount of time that would be biologically interesting).

Two system states differing in only a small number of state variables (i.e., having only a few bits that differ out

of the entire set of N variables) often lie on dynamical trajectories that converge closer to one another in state

space. In other words, their attractors are robust under small perturbations. However, there can be states

within a basin of attraction that differ in only one state variable from a trajectory that can lead to a different

attractor.

(^76) This point is discussed further in Section 5.4.2.2 and the references therein.
(^77) Z. Szallasi and S. Liang, “Modeling the Normal and Neoplastic Cell Cycle with ‘Realistic Boolean Genetic Networks’: Their
Application for Understanding Carcinogenesis and Assessing Therapeutic Strategies,” Pacific Symposium on Biocomputing, pp. 66-
76, 1998.