Finding Cycles in Synchronous Boolean Networks With Applications to Biochemical Systems

Abstract

This paper is an analytical study of Boolean networks. The motivation is our desire to understand the large, complicated and interconnected pathways which comprise intracellular biochemical signal transduction networks. The simplest possible conceptual model that mimics signal transduction with sigmoidal kinetics is the n-node Boolean network each of whose elements or nodes has the value 0 (off) or 1 (on) at any given time T = 0, 1, 2, …. A Boolean network has 2nstates all of which are either on periodic cycles (including fixed points) or transients leading to cycles. Thus one understands a Boolean network by determining the number and length of its cycles. The problem one must circumvent is the large number of states (2n) since the networks we are interested in have 100 or more elements. Thus we concentrate on developing size n methods rather than the impossible task of enumerating all 2nstates. This is done as follows: the dynamics of the network can be described by n polynomial equations which describe the logical function which determines the interaction at each node. Iterating the equations one step at a time finds all fixed points, period two cycles, period three cycles, etc. This is a general method that can be used to determine the fixed points and moderately large periodic cycles of any size network, but it is not useful in finding the largest cycles in a large network. However, we also show that the network equations can often be reduced to scalar form, which makes the cycle structure much more transparent. The scalar equations method is a true "size n" method and several examples (including nontrivial biochemical systems) are examined.

An efficient algorithm for computing fixed length attractors based on bounded model checking in synchronous Boolean networks with biochemical applications

“…All experiments can be performed on an Intel Core TM CPU 4300 1.80 GHz with 2 GB memory on an Ubuntu 9.04 Linux server. The top 4 Boolean networks were generated from 5 organisms, including Arabidopsis thaliana (Chaos et al, 2006), budding yeast (Li et al, 2004), Drosophila melanogaster (Albert and Othmer, 2003), T-helper cell (Mendoza and Xenarios, 2006), and protein (Gonze and Goldbeter, 2001;Heidel et al, 2003). In addition to these 5 classical benchmarks, we provide the 2 other randomly generated Boolean networks.…”

“…Based on previous studies (Glass and Kaufmann, 1973;Heidel et al, 2003), one condition of sigmoidal kinetics systems was chosen from a complicated environment, which is shown using the following differential equations:…”

“…Based on a study of Heidel et al (2003), we simplified the above differential equations as the Boolean network shown in Figure 1 (Boolean network) and its Boolean translation function. This is a nonlinear logic function with negative feedback.…”

An efficient algorithm for finding attractors in synchronous Boolean networks with biochemical applications

“…We use two realistic biochemical system examples from previous reports (Robert, 1986;Heidel et al, 2003) to analyze the conditions of the SCAs. To begin, we use a simple affine system (line terms and constant terms) (Milligan and Wilson, 1993;Heidel et al, 2003) as an example.…”

“…Recently, Heidel (Heidel et al, 2003) and Farrow (Farrow et al, 2004) found the SCAs in synchronous Boolean networks. Zhao (Zhao, 2005) also proved the computing SCAs' methods in synchronous Boolean networks to be a nondeterministic polynomial time (NP) complete problem.…”