Evolving Boolean Regulatory Networks with Variable Gene Expression Times

📝 Abstract

The time taken for gene expression varies not least because proteins vary in length considerably. This paper uses an abstract, tuneable Boolean regulatory network model to explore gene expression time variation. In particular, it is shown how non-uniform expression times can emerge under certain conditions through simulated evolution. That is, gene expression time variance appears beneficial in the shaping of the dynamical behaviour of the regulatory network without explicit consideration of protein function.

💡 Analysis

The time taken for gene expression varies not least because proteins vary in length considerably. This paper uses an abstract, tuneable Boolean regulatory network model to explore gene expression time variation. In particular, it is shown how non-uniform expression times can emerge under certain conditions through simulated evolution. That is, gene expression time variance appears beneficial in the shaping of the dynamical behaviour of the regulatory network without explicit consideration of protein function.

📄 Content

Evolving Boolean Regulatory Networks with Variable Gene Expression Times
Larry Bull Department of Computer Science & Creative Technologies,
University of the West of England, Bristol BS16 1QY, U.K.

Larry.Bull@uwe.ac.uk Abstract. The time taken for gene expression varies not least because proteins vary in length considerably. This paper uses an abstract, tuneable Boolean regulatory network model to explore gene expression time variation. In particular, it is shown how non-uniform expression times can emerge under certain conditions through simulated evolution. That is, gene expression time variance can be beneficial in the shaping of the dynamical behaviour of the regulatory network without explicit consideration of protein function.

1. Introduction A protein’s function is dependent upon its tertiary (3D) structure which in turn is dependent upon the primary structure of the amino acid sequence by which it is specified. Typically, the more amino acids in the primary structure, the more complex the tertiary structure. Similarly, the more amino acids, the longer gene expression can be expected to take. The lengths of genes/amino acid sequences varies considerably within and across taxa. It is well-established that, due to chemical equivalences between amino acid sequences, there is a strong neutrality effect at the molecular level of evolution (eg, [11]). However, this does not fully explain the differences in the distribution of protein lengths seen, nor why eukaryotic proteins are typically larger than bacterial proteins (eg, [15]).
   With the aim of enabling the systematic exploration of artificial genetic regulatory network models (GRN), a simple approach to combining them with abstract fitness landscapes has been presented [2]. More specifically, random Boolean networks (RBN) [8] were combined with the NK model of fitness landscapes [10]. In the combined form – termed the RBNK model – a simple relationship between the states of N randomly assigned nodes within an RBN is assumed such that their value is used within a given NK fitness landscape of trait dependencies. This paper explores the introduction of variable expression times to the genes in a traditional Boolean regulatory network within the RBNK model. That is, the effects of protein length variation are considered based purely upon the dynamical behaviour of the regulatory network being shaped under an evolutionary process. It is shown that non-uniform gene expression times can be selected for and that a relationship appears to exist between gene length and gene connectivity in such cases.
2. The RBNK Model Within the traditional form of RBN, a network of R nodes, each with a randomly assigned Boolean update function and B directed connections randomly assigned from other nodes in the network, all update synchronously based upon the current state of those B nodes (Figure 1). Hence those B nodes are seen to have a regulatory effect upon the given node, specified by the given Boolean function attributed to it. Since they have a finite number of possible states and they are deterministic, such networks eventually fall into an attractor. It is well-established that the value of B affects the emergent behaviour of RBN wherein attractors typically contain an increasing number of states with increasing B (see [9] for an overview). Three regimes of behaviour exist: ordered when B=1, with attractors consisting of one or a few states; chaotic when B≥3, with a very large number of states per attractor; and, a critical regime around B=2, where similar states lie on trajectories that tend to neither diverge nor converge (see [3] for formal analysis). Note that traditionally the size of an RBN is labelled N, as opposed to R here, and the degree of node connectivity labelled K, as opposed to B here. The change is adopted due to the traditional use of the labels N and K in the NK model of fitness landscapes which are also used in this paper, as will be shown. Kauffman and Levin [10] introduced the NK model to allow the systematic study of various aspects of fitness landscapes (see [9] for an overview). In the standard NK model an individual is represented by a set of N (binary) genes or traits, each of which depends upon its own value and that of K randomly chosen others in the individual (Figure 1). Thus increasing K, with respect to N, increases the epistasis. This increases the ruggedness of the fitness landscapes by increasing the number of fitness peaks. The NK model assumes all epistatic interactions are so complex that it is only appropriate to assign (uniform) random values to their effects on fitness. Therefore for each of the possible K interactions, a table of 2(K+1) fitnesses is created, with all entries in the range 0.0 to 1.0, such that there is one fitness value for each combination of traits. The fitness contribution of each trait is found from its individual table. These fi

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