Towards Optimal Decomposition of Boolean Networks

Temporary and permanent control of partially specified Boolean networks

Boolean networks (BNs) are a well-accepted modelling formalism in computational systems biology. Nevertheless, modellers often cannot identify only a single BN that matches the biological reality. The typical reasons for this is insufficient knowledge or a lack of experimental data. Formally, this uncertainty can be expressed using partially specified Boolean networks (PSBNs), which encode the wide range of network candidates into a single structure. In this paper, we target the control of PSBNs. The goal of BN control is to find perturbations which guarantee stabilisation of the system in the desired state. Specifically, we consider variable perturbations (gene knock-out and over-expression) with three types of application time-window: one-step, temporary, and permanent. While the control of fully specified BNs is a thoroughly explored topic, control of PSBNs introduces additional challenges that we address in this paper. In particular, the unspecified components of the model cause a significant amount of additional state space explosion. To address this issue, we propose a fully symbolic methodology that can represent the numerous system variants in a compact form. In fully specified models, the efficiency of a perturbation is characterised by the count of perturbed variables (the perturbation size). However, in the case of a PSBN, a perturbation might work only for a subset of concrete BN models. To that end, we introduce and quantify perturbation robustness. This metric characterises how efficient the given perturbation is with respect to the model uncertainty. Finally, we evaluate the novel control methods using non-trivial real-world PSBN models. We inspect the method's scalability and efficiency with respect to the size of the state space and the number of unspecified components. We also compare the robustness metrics for all three perturbation types. Our experiments support the hypothesis that one-step perturbations are significantly less robust than temporary and permanent ones.

An Efficient Approach Towards the Source-Target Control of Boolean Networks

We study the problem of computing a minimal subset of nodes of a given asynchronous Boolean network that need to be perturbed in a single-step to drive its dynamics from an initial state to a target steady state (or attractor), which we call the source-target control of Boolean networks. Due to the phenomenon of state-space explosion, a simple global approach that performs computations on the entire network may not scale well for large networks. We believe that efficient algorithms for such networks must exploit the structure of the networks together with their dynamics. Taking this view, we derive a decomposition-based solution to the minimal source-target control problem which can be significantly faster than the existing approaches on large networks. We then show that the solution can be further optimized if we take into account appropriate information about the source state. We apply our solutions to both real-life biological networks and randomly generated networks, demonstrating the efficiency and efficacy of our approach.