Elsevier

Biosystems

Volume 98, Issue 3, December 2009, Pages 127-136
Biosystems

A multi-objective differential evolutionary approach toward more stable gene regulatory networks

https://doi.org/10.1016/j.biosystems.2009.09.002Get rights and content

Abstract

Models are of central importance in many scientific contexts. Mathematical and computational modeling of genetic regulatory networks promises to uncover the fundamental principles of living systems. Biological models, such as gene regulatory models, can help us better understand interactions among genes and how cells regulate their production of proteins and enzymes. One feature shared among living systems is their ability to cope with perturbations and remain stable, a property that is the result of evolutionary fine-tuning over many generations. In this study we use random Boolean networks (RBNs) as an abstract model of gene regulatory systems. By applying Differential Evolution (DE), an evolution-based optimization technique, we produce networks with increased stability. DE requires relatively few user-specified parameters, has fast convergence and does not rely on initial conditions to find the global minima within multi-dimensional search spaces.

The stability of networks is evaluated by taking parameters such as their network sensitivity, attractor cycle length and number of attractor basins into account. In this study we present an evolutionary approach to produce networks with specific properties (high stability) starting from chaotic regimes. From this chaotic regime and randomly generated classical RBNs with static input degree, our evolutionary approach is able to produce networks with homogenous Boolean functions and highly structured state spaces.

Introduction

With new advancements in genomics and developmental biology, an enormous amount of genetic data is produced each day. The genetic data is used to understand complex and dynamic behaviors of genetic regulatory systems. Critical cell functions are carried out through many genetic interactions. Understanding the regulation processes is crucial for identifying cellular responses to internal and external signals. This is because cells respond to internal and external signals by switching different genes on and off, so that for example, proteins are manufactured in response to signaling molecules. Modeling genetic networks helps analyze genetic interactions and controls. These artificial genetic networks give insight into how genes regulate the production of proteins in cells and ultimately how cellular responses arise. This has applications in controlling cell differentiation (replication of the DNA preceding cell division) and the unfolding of developmental processes. Genes and their expression also play an important role in the stability of cancer cells. Hence, understanding the fundamental aspect of gene regulation through mathematical and computational modeling provides a framework to understand cancer at its most fundamental level (Boulton et al., 2002).

Random Boolean networks (RBNs) have attracted attention as abstract models to study gene regulatory networks and their time dynamics (Kauffman, 1993, Kauffman, 2004). RBN-based approaches have also been used to investigate and classify cellular automata (Wolfram, 1994, Wolfram, 2002), neural networks (Derrida et al., 1987, Hopfield, 1982) and spin glasses (Anderson, 1983). Through mathematical and statistical models a wide range of RBN properties have been studied such as their topological compositions (Kauffman, 1993, Kauffman, 2004, Aldana, 2003, Iguchi et al., 2007, Matache and Heidel, 2004), and what constitutes biologically meaningful regulatory functions (Raeymaekers, 2001). How random Boolean networks tend to react to perturbations (Shmulevich and Kauffman, 2004) is related to the number and length of their state space attractors (Drossel et al., 2005, Somogyvari et al., 2000). In this same context of stability analysis, RBN criticality and scaling properties have been investigated (Derrida and Stauffer, 1986, Mihaljev and Drossel, 2006). Although most analytical research is performed on smaller size networks, simulations of large RBNs have been undertaken as well (Hawick and Scogings, 2007).

The focus of this paper is on autonomous, classical random Boolean networks. Boolean networks with no inputs from outside the system are considered to be autonomous. We study ensembles of RBNs, that is groups of regulatory networks that share common properties regarding their stability and other measurable features that would make these networks akin to gene regulatory systems in real cells (Kauffman, 2004). Focusing on single objective optimization, evolution of RBNs and their adaptability to perturbations have been investigated before (Lemke et al., 2001, Frank, 1999, Mihaljev and Drossel, 2009, Bornholdt and Sneppen, 1998). In our study we investigate the behavior of gene regulatory networks (GRNs) modeled by RBNs and focus on the evolvability of RBN ensembles while monitoring their corresponding network properties. The focus of this work is to use heuristic optimization in order to produce more stable and robust networks that maintain their functionality under perturbation (Boccalettia et al., 2006). In order to evolve the RBN ensembles we take a multi-objective approach to tune networks toward higher stability.

The rest of our paper is organized as follows. Section 2 gives formal definitions of random Boolean networks and their associated state spaces. In Section 3 we briefly outline how to classify RBN dynamics. How we evaluate RBN fitness and evolve them is presented in Section 4. Our simulation results are presented in Section 5 and in Section 6 we analyze our results. Finally, in Section 7 we conclude the paper and discuss our future investigations into the evolution of RBN ensembles.

Section snippets

Random Boolean networks (RBNs)

We define a random Boolean network RBN(N,K)=(G,F) as a graph, G, together with a set of Boolean functions, F. The parameters N and K are positive integers that define the number of nodes and the in-degree for each node, respectively. The graph represents the set of genes and how some of their products – those that act as transcription factors – regulate other genes. Hence, the transcription network G=(G,E) is defined by its nodes, G={g1,,gN}, which constitute the set of genes, and regulatory,

Analysis of RBN dynamics

One key feature of an RBN is its ability to stabilize or ‘settle’ after an initial period of activity. Since there is a finite number of states (2N) in the state space of an RBN, eventually a given state must be repeated. These state cycles are referred to as attractors (Kauffman, 1993). Point attractors, the simplest kind of attractors, have a cycle length of one, where a state is its own successor state. The set of states that flow into an attractor is called the basin of attraction.

Evolution of Boolean networks

Differential Evolution (DE), introduced by Storn and Price (Storn and Price, 1997), is a stochastic, population-based optimization algorithm. In DE a population of solution vectors is updated by matrix addition, and subtraction and by swapping components until the population converges to the optimum point of interest. One of the advantages of DE over standard evolutionary algorithms (Esmaeili and Jacob, 2008a) is that DE requires relatively few user-specified parameters, has fast convergence,

Results

In this study we have evolved RBNs of size 8 and input degree 6 (RBN(8,6)) over 100 generations. Only the initial population exhibits a uniform input degree. We impose no restriction on the input degree of later generations. Each generation contains 20 individuals. In a given evolutionary run we monitor 2000 networks. We have replicated our experiments over 100 trials and examined a total of 200,000 networks. We have chosen size 8 and input degree 6 to start the system; that is we begin with

Discussion

Starting from a population of networks in chaotic regimes (K=6), the evolutionary process presented here is able to obtain networks which exhibit frozen regimes with low input degree. Analyzing the topology of networks generated by the evolutionary process, on average, the input degree to each gene in a given network promoted by the fitness function is two. Almost all genes in a given network either have an input degree of 2 or 4.

The number of attractors and attractor cycle lengths of Kauffman

Conclusion

With the aid of Differential Evolution we have demonstrated an efficient search procedure through the space of random Boolean networks. The DE algorithm used converges very quickly and requires relatively few user-specified constraints and parameters. This allows the simulation to deal with larger population sizes compared to previous work in evolutionary RBNs (Esmaeili and Jacob, 2008a). Results also show a decrease in sensitivity, in attractor cycle length and in the number of attractors. The

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