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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 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 to stabilise the system in the desired phenotype. We allow the specification of the Boolean model representing a given RN to be incomplete. To that end, we utilise the formalism of partially specified Boolean networks which covers every possible behaviour of unspecified parts of the system. Such an approach causes a significant state explosion. This problem is addressed by using symbolic methods to represent both the unspecified model parts and all possible perturbations of the system. Additionally, to make the control design efficient and practically applicable, the optimal control should be minimal in terms of size. Moreover, in a partially specified model, a control may achieve the desired stabilisation only for a subset of the possible fully specified model instantiations. To address these aspects, we utilise several quantitative measures. Apart from the size of perturbation, we also examine its robustness -- a portion of instantiations for which the control is applicable. We show that proposed symbolic methods solving the control problem for partially specified BNs are efficient and scale well. 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.

This technical report relates Biochemical Space Language (BCSL) to Multiset rewriting systems (MRS). For a BCSL model, the semantics are defined in terms of transition systems, while for an MRS, they are defined in terms of a set of runs. In this report, we relate BCSL to MRS by first showing how the transition system is related to a set of runs and consequently showing how for every BCSL model, an MRS can be constructed such that both represent the same set of runs. The motivation of this step is to establish BCSL in the context of a more general rewriting system and benefit from properties shown for them. Finally, we show that regulations defined for MRS can be consequently used in the BCSL model., Comment: 9 pages, 8 figures

Multiset rewriting systems provide a formalism particularly suitable for the description of biological systems. We present an extension of this formalism with additional controls on the derivations as a tool for reducing possible non-deterministic behaviour by providing additional knowledge about the system. We introduce several regulation mechanisms and compare their generative power., Comment: 12 pages, 14 figures

Edge-coloured directed graphs provide an essential structure for modelling and analysis of complex systems arising in many scientific disciplines (e.g. feature-oriented systems, gene regulatory networks, etc.). One of the fundamental problems for edge-coloured graphs is the detection of strongly connected components, or SCCs. The size of edge-coloured graphs appearing in practice can be enormous both in the number of vertices and colours. The large number of vertices prevents us from analysing such graphs using explicit SCC detection algorithms, such as Tarjan's, which motivates the use of a symbolic approach. However, the large number of colours also renders existing symbolic SCC detection algorithms impractical. This paper proposes a novel algorithm that symbolically computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs $O(p \cdot n \cdot log~n)$ symbolic steps, where $p$ is the number of colours and $n$ is the number of vertices. We evaluate the algorithm using an experimental implementation based on binary decision diagrams (BDDs). Specifically, we use our implementation to explore the SCCs of a large collection of coloured graphs (up to $2^{48}$) obtained from Boolean networks -- a modelling framework commonly appearing in systems biology.

Boolean network (BN) is a simple model widely used to study complex dynamic behaviour of biological systems. Nonetheless, it might be difficult to gather enough data to precisely capture the behavior of a biological system into a set of Boolean functions. These issues can be dealt with to some extent using parametrised Boolean networks (ParBNs), as it allows to leave some update functions unspecified. In this paper, we attack the control problem for ParBNs with asynchronous semantics. While there is an extensive work on controlling BNs without parameters, the problem of control for ParBNs has not been in fact addressed yet. The goal of control is to ensure the stabilisation of a system in a given state using as few interventions as possible. There are many ways to control BN dynamics. Here, we consider the one-step approach in which the system is instantaneously perturbed out of its actual state. A naive approach to handle control of ParBNs is using parameter scan and solve the control problem for each parameter valuation separately using known techniques for non-parametrised BNs. This approach is however highly inefficient as the parameter space of ParBNs grows doubly-exponentially in the worst case. In this paper, we propose a novel semi-symbolic algorithm for the one-step control problem of ParBNs, that builds on a symbolic data structures to avoid scanning individual parameters. We evaluate the performance of our approach on real biological models.

We present the second generation of a rule-based language called Biochemical Space Language (BCSL) that combines the advantages of different approaches and thus makes an effort to overcome several problems with existing solutions. The key aspect of the language is the level of abstraction it uses, which allows scalable and compact hierarchical specification of biochemical entities. This abstraction enables unique analysis techniques to reason about properties of models written in the language on the semantic and syntactic level., Comment: 18 pages, 5 figures

The modelling of discrete regulatory networks combines a graph specifying the pairwise influences between the variables of the system, and a parametrisation from which can be derived a discrete transition system. Given the influence graph only, the exploration of admissible parametrisations and the behaviours they enable is computationally demanding due to the combinatorial explosions of both parametrisation and reachable state space. This article introduces an abstraction of the parametrisation space and its refinement to account for the existence of given transitions, and for constraints on the sign and observability of influences. The abstraction uses a convex sublattice containing the concrete parametrisation space specified by its infimum and supremum parametrisations. It is shown that the computed abstractions are optimal, i.e., no smaller convex sublattice exists. Although the abstraction may introduce over-approximation, it has been proven to be conservative with respect to reachability of states. Then, an unfolding semantics for Parametric Regulatory Networks is defined, taking advantage of concurrency between transitions to provide a compact representation of reachable transitions. A prototype implementation is provided: it has been applied to several examples of Boolean and multi-valued networks, showing its tractability for networks with numerous components., Comment: preprint

This report proposes a novel framework for a rigorous robustness analysis of stochastic biochemical systems. The technique is based on probabilistic model checking. We adapt the general definition of robustness introduced by Kitano to the class of stochastic systems modelled as continuous time Markov Chains in order to extensively analyse and compare robustness of biological models with uncertain parameters. The framework utilises novel computational methods that enable to effectively evaluate the robustness of models with respect to quantitative temporal properties and parameters such as reaction rate constants and initial conditions. The framework is applied to gene regulation as an example of a central biological mechanism where intrinsic and extrinsic stochasticity plays crucial role due to low numbers of DNA and RNA molecules. Using our methods we have obtained a comprehensive and precise analysis of stochastic dynamics under parameter uncertainty. Furthermore, we apply our framework to compare several variants of two-component signalling networks from the perspective of robustness with respect to intrinsic noise caused by low populations of signalling components. We succeeded to extend previous studies performed on deterministic models (ODE) and show that stochasticity may significantly affect obtained predictions. Our case studies demonstrate that the framework can provide deeper insight into the role of key parameters in maintaining the system functionality and thus it significantly contributes to formal methods in computational systems biology., Comment: 43 pages, 15 figures, technical report

To express temporal properties of dense-time real-valued signals, the Signal Temporal Logic (STL) has been defined by Maler et al. The work presented a monitoring algorithm deciding the satisfiability of STL formulae on finite discrete samples of continuous signals. The logic has been used to express and analyse biological systems, but it is not expressive enough to sufficiently distinguish oscillatory properties important in biology. In this paper we define the extended logic STL* in which STL is augmented with a signal-value freezing operator allowing us to express (and distinguish) detailed properties of biological oscillations. The logic is supported by a monitoring algorithm prototyped in Matlab. The monitoring procedure of STL* is evaluated on a biologically-relevant case study., Comment: In Proceedings HSB 2012, arXiv:1208.3151

In this paper a novel computational technique for finite discrete approximation of continuous dynamical systems suitable for a significant class of biochemical dynamical systems is introduced. The method is parameterized in order to affect the imposed level of approximation provided that with increasing parameter value the approximation converges to the original continuous system. By employing this approximation technique, we present algorithms solving the reachability problem for biochemical dynamical systems. The presented method and algorithms are evaluated on several exemplary biological models and on a real case study. This is a full version of the paper published in the proceedings of CompMod 2011., Comment: Full version of a paper accepted to COMPMOD 2011

In this paper a novel tool BioDiVinEfor parallel analysis of biological models is presented. The tool allows analysis of biological models specified in terms of a set of chemical reactions. Chemical reactions are transformed into a system of multi-affine differential equations. BioDiVinE employs techniques for finite discrete abstraction of the continuous state space. At that level, parallel analysis algorithms based on model checking are provided. In the paper, the key tool features are described and their application is demonstrated by means of a case study.