Boolean models

If the system of interest has already been studied and chemical or kinetic parameters are determined, a common approach is to make use of these details withinquantitative differential equation models with continuous protein con-centration values (Schoeberl et al., 2002; Hatakeyama et al., 2003; Jones et al., 2006; Chen et al., 2009). Lacking such detailed information, less complex but more abstractqualitative Boolean models can be applied. Here, the activity states of the network components, i.e. genes ore proteins, are modelled in a discrete way as active (1) or inactive (0) (Sahin et al., 2009; Samaga et al., 2009). This simplification is especially attractive for computationally extensive large-scale systems (Wang et al., 2012).

Wang et al. (2012) provide a methodology overview of Boolean modelling in systems biology. An excerpt, supplemented by information from the vignette of the BoolNet (M¨ussel et al., 2010) R package which provides analysis tools for Boolean networks, is described in the following. A Boolean network consists of a set of nodes with binary states which are determined by other network nodes through Boolean functions. For example, protein P₁ is activated by phospho-rylation by proteinP₂ orP₃ and inhibited by protein P₄. Such dependencies betweennvariables are expressed via Boolean functions which map the set of all n-tuples over {0,1} to a binary output, i.e. {0,1}ⁿ → {0,1}. In the mentioned example n equals four. Boolean functions are based on logical operations via AND, OR and NOT. In the example the corresponding function to determine the activity state of proteinP₁would beP₁ = (P₁ ORP₂ ORP₃) AND NOTP₄. In general, a Boolean network model consists ofnBoolean variables{σ₁, σ₂, ..., σ_n} each assigned with a Boolean functionB ={B₁, B₂, ..., B_n} which determines

14 Introduction

the value of the variable. Hence, the state of the network at time t can be represented by the state vector σ(t) = (σ₁(t), σ₂(t), ..., σ_n(t)). The so-called state space contains all possible 2ⁿ states.

Within thesynchronous scheme, the subsequent state of the network σ(t+ 1) is computed by applying all transition functionsB_i(σ(t)), i= 1,2, ..., n, simul-taneously. Biologically this means that all genes are updated at the same time.

Within the asynchronous scheme, it is assumed that gene expression levels are likely to change at different time points. A common stochastic asynchronous scheme looks like the following. At each time point t only one transition functionBi ∈B is chosen randomly and the corresponding variable is updated.

In probabilistic Boolean networks (PBNs) more than one transition function can be specified per variable. Each function is chosen with a specific probability, and the probabilities of all functions for one variable sum up to one. A state transition is performed by choosing one function for each variable according to the probabilities and by applying the chosen functions synchronously.

State transition graphs represent all possible trajectories in the state space.

The corresponding nodes represent the system states and the edges represent the state transitions which are allowed according to the updating scheme. The subsequent state updates at each time step form a trajectory of states which either leads to a steady state, which remains unchanged under further system updates, or a set of recurring states. Such states are called attractors and represent the stable states of cell function in which the network mostly resides.

Consequently, they reflect system phenotypes under specific perturbations. All states that lead to a specific attractor form its so-called basin of attraction.

Apart from steady-state attractors, simple and complex attractors exist. Simple attractors occur in synchronous Boolean networks and consist of a set of states among which the system oscillates in a cycle. Complex or loose attractors occur in asynchronous networks and consist of a set of states among which the system oscillates irregularly. A state of this set can be reached from all other states in the set. Wang et al. (2012) refer to several approaches for attractor calculations. As the R packageBoolNet was applied in the modelling study of this dissertation, the following description is based on the methods of this software. Within an exhaustive synchronous search, the software starts from all possible network states and conducts synchronous state transitions until a simple or steady-state attractor is reached. This approach is rather

1.2 Theoretical background and prior research 15 time and memory consuming with increasing network size. BoolNet supports this method just for networks up to 29 genes due to memory restrictions in R. However, BoolNet offers synchronous and asynchronous heuristic search algorithms as well, which start from a subset of predefined or random states.

The asynchronous heuristic performs random transitions leading to an attractor with high probability which is afterwards tested for being complex. Finally, Markov chain simulations can be used to identify relevant states by calculating the probability to reach a state after a predefined number of iterations. If the number is large enough, it is very likely to reach an attractor.

A Boolean network can be expressed as a directed graph G(V, E) in which the nodes V = {v₁, v₂, ..., v_n} represent Boolean variables and the edge set E is defined by Boolean functions. Edges can be activating or inhibiting according to the node effects in the Boolean functions. Figure 1.4 shows an exemplary Boolean network (a), related Boolean functions (b), truth tables mapping input to output values (c) and the synchronous state transition graph (d).

FIGURE 1.4. Example of a simple Boolean network model (Wang et al. (2012),©IOP Publishing. Reproduced by permission of IOP Publishing. All rights reserved.).

16 Introduction

The R package BoolNet provides tools for assembly, analysis and visualization of Boolean networks. Apart from synchronous, asynchronous and probabilistic Boolean networks, so-called temporal Boolean networks are supported. Here, the subsequent state σ(t+ 1) may not only depend on the direct preceding stateσ(t) but on any predecessor state or the time step itself. The software offers different methods for network assembly. Networks can be predefined and loaded from files, reconstructed from time course data or generated randomly.

Furthermore, attractors can be calculated and perturbation simulations can be conducted including overexpressed or knock-out genes, for example, to test the robustness of networks to noise and mismeasurements. Above that, visualization tools are offered. Boolean networks can be graphically represented as graphs in which genes correspond to the vertices and the inputs defined by the transition functions determine the edges. Transition states can be visualized in table format with row-wise representation of the genes and column-wise representation of the consecutive states until the attractor is reached. One can also generate a transition graph of the network nodes and state transitions in which attractors are highlighted in bold and different colours allow to distinguish the basins of attraction. Simple attractors and the corresponding set of states can be visualized as well in table format. Also a graph representation of complex attractors is offered with nodes representing the related states and arrows representing the state transitions.

Several import and export interfaces exist, but as the R packageddepn (Bender et al., 2010, 2011) was chosen within this dissertation for network reconstruction, it was especially extended to convert the resulting adjacency matrices to logical rules which can be used as input forBoolNet.