Dynamical properties of model gene networks and implications for the inverse problem

Abstract

We study the inverse problem, or the “reverse-engineering” problem, for two abstract models of gene expression dynamics, discrete-time Boolean networks and continuous-time switching networks. Formally, the inverse problem is similar for both types of networks. For each gene, its regulators and its Boolean dynamics function must be identified. However, differences in the dynamical properties of these two types of networks affect the amount of data that is necessary for solving the inverse problem. We derive estimates for the average amounts of time series data required to solve the inverse problem for randomly generated Boolean and continuous-time switching networks. We also derive a lower bound on the amount of data needed that holds for both types of networks. We find that the amount of data required is logarithmic in the number of genes for Boolean networks, matching the general lower bound and previous theory, but are superlinear in the number of genes for continuous-time switching networks. We also find that the amount of data needed scales as 2^K, where K is the number of regulators per gene, rather than $2^{2K}$, as previous theory suggests.

Introduction

Abstract models of genetic networks, such as Boolean networks and continuous-time switching networks, have been proposed as conceptual models for helping us understand the behavior of real genetic networks (Glass, 1975, Kauffman, 1969, Kauffman, 1993). These formalisms, or generalizations of them, have been used to model such systems as the sporulation network in Bacillus subtilis (de Jong et al., 2004a), the gap gene network of Drosophila melanogaster (Sanchez and Thieffry, 2001), and the segment polarity network in the same organism (Albert and Othmer, 2003). Although these models omit many of the details of the real chemical interactions, they serve as useful syntheses of the often-distributed biological knowledge about these networks, they allow testing of hypotheses about network behavior, and they sometimes lead to new biological hypotheses.

General properties of model networks, particularly randomly generated networks, have been studied in an effort to understand basic principles behind the functioning of real networks. For example, Kauffman has equated different cell types in real organisms with different fixed points or cycles in the dynamics of model networks (Kauffman, 1969, Kauffman, 1993). He and others have studied how the number and period of attractors in random networks depends on network size, topology, and how the dynamics functions are chosen (Bagley and Glass, 1996, Bastolla and Parisi, 1997, Bilke and Sjunnesson, 2001, Glass and Hill, 1998, Kauffman, 1969, Kauffman, 1993, Kauffman et al., 2003, Raeymaekers, 2002, Samuelsson and Troein, 2003, Shmulevich and Kauffman, 2004, Socolar and Kauffman, 2003).

The increasing availability of quantitative gene expression data has kindled the hope of automatically inferring regulatory relationships in real gene networks. There have been some successes (e.g. Jaeger et al., 2004a, Jaeger et al., 2004b, Reinitz and Sharp, 1995), but there is not yet a standard methodology for doing so. Much remains to be understood about the problem. What is the computational complexity of the problem? What algorithms work best? How much data is needed? How should data be collected? Are there fundamental limits on what can be inferred from expression data alone?

Analyses of the inverse problem for Boolean and continuous-time switching networks have begun to provide theoretical answers to these questions. Liang et al. (1998) were the first to propose a solution to the inverse problem for Boolean networks. Later, Akutsu et al. (1999) and Ideker et al. (2000) described alternative solutions. Perkins et al. (2004) described solutions to the inverse problem for continuous-time switching networks. The approaches proposed for Boolean networks can also be applied to continuous-time switching networks, though the methods of Liang et al. (1998) and Akutsu et al. (1999) in particular require significantly more computation than the method described in Perkins et al. (2004).

We focus on the sample complexity of the inverse problem—that is, how much data is needed to identify the network? In particular, we study the problem for Boolean and continuous-time switching networks of N genes in which each gene has precisely K regulators. Akutsu et al. (1999) studied this problem for Boolean networks under the assumption that the data comprises uniformly randomly sampled states of the network. They proved that the amount of data needed scales as log N and as $2^{2K}$. The $2^{2K}$ term is disheartening, because it suggests that it will take enormous amounts of data to identify densely connected networks. However, the log N dependence is encouraging because it suggests that network size per se is not a very important factor.

We consider solving the inverse problem based on time series data, although, as we argue in Section 4, time series data from randomly generated Boolean networks behave in many respects as randomly sampled data. In Section 5, we show that, regardless of how the data is generated, solving the inverse problem for Boolean networks or continuous-time switching networks requires at least $\frac{1}{2}(2^K+K(\text{lo}{\text{g}}_2(N-K)-\text{lo}{\text{g}}_{2}K))$ samples. We then derive new estimates for the expected amount of data required. It turns out that differences in the dynamical properties of these networks, examined in Section 4, have a significant impact. In Section 5, we estimate the expected sample complexity for Boolean networks as O(K2^K log N) and for continuous-time switching networks as O(2^KN log N). These estimates are supported by simulation experiments, reported in Section 6.