Robust Control of Partially Specified Boolean Networks

Regulatory networks (RNs) are a well-accepted modelling formalism in computational systems biology. The control of RNs is currently receiving a lot of attention because it provides a theoretical and computational basis for cell reprogramming – an attractive technology developed in regenerative medicine. By solving the control problem, we learn which parts of a biological system should be perturbed in order for it to stabilise the system in the desired phenotype. In this paper, we use Boolean networks (BNs) with asynchronous update to represent the dynamics of an RN as a discrete finite-state system. Furthermore, we allow the specification of the Boolean model representing a given RN to be incomplete (partial). This is crucial in cases where the exact Boolean update functions of the BN are not fully known (e.g. due to a lack of measured data). To that end, we utilise the formalism of partially specified Boolean networks which allows us to cover every possible behaviour reflecting the unspecified parts of the system. Such an approach inevitably causes a significant state explosion. This problem is efficiently addressed by using symbolic methods to represent both the unspecified model behaviour as well as all possible perturbations of the system. Our framework supports several ways to control the system by employing one-step, temporary, and permanent perturbations. Additionally, to make the control design efficient and practically applicable, the optimal control should be minimal in terms of the size – the number of perturbed system components. Moreover, in a partially specified model, a control may achieve the desired stabilisation only for a subset of the possible fully specified instantiations of the model. To address these aspects in the control design, we utilise several quantitative measures. In particular, apart from the size of perturbation necessary to achieve control, we also examine its robustness – a portion of fully specified models for which the control is applicable. We show that the proposed symbolic methods solving the control problem for partially specified BNs are efficient and scale well with the number of unspecified model elements. The provided experiments demonstrate that our algorithms can handle relatively large BNs. We also evaluate the robustness metrics in cases of all three studied control types. The robustness metric tells us how big a proportion of fully defined systems the given perturbation works. Our experiments support the hypothesis that one-step perturbations may be less robust than temporary or permanent perturbations. This is a full version of a paper that is submitted to a journal.