Dynamical and Structural Modularity of Discrete Regulatory Networks

A biological regulatory network can be modeled as a discrete function that contains all available information on network component interactions. From this function we can derive a graph representation of the network structure as well as of the dynamics of the system. In this paper we introduce a method to identify modules of the network that allow us to construct the behavior of the given function from the dynamics of the modules. Here, it proves useful to distinguish between dynamical and structural modules, and to define network modules combining aspects of both. As a key concept we establish the notion of symbolic steady state, which basically represents a set of states where the behavior of the given function is in some sense predictable, and which gives rise to suitable network modules. We apply the method to a regulatory network involved in T helper cell differentiation

Symbolic Steady States and Dynamically Essential Subnetworks of Discrete Regulatory Networks

Analyzing complex networks is a difficult task, regardless of the chosen modeling framework. For a discrete regulatory network, even if the number of components is in some sense manageable, we have to deal with the problem of analyzing the dynamics in an exponentially large state space. A well known idea to approach this difficulty is to identify smaller building blocks of the system the study of which in isolation still renders information on the dynamics of the whole network. In this talk, we introduce the notion of symbolic steady state which allows us to identify such building blocks. We state explicit rules how to derive attractors of the network from subnetwork attractors valid for synchronous as well as asynchronous dynamics. Illustrating those rules, we derive general conditions for circuits embedded in the network to transfer their behavioral characteristics pertaining number and size of attractors observed in isolation to the complex network

Basins of Attraction, Commitment Sets and Phenotypes of Boolean Networks

The attractors of Boolean networks and their basins have been shown to be highly relevant for model validation and predictive modelling, e.g., in systems biology. Yet there are currently very few tools available that are able to compute and visualise not only attractors but also their basins. In the realm of asynchronous, non-deterministic modeling not only is the repertoire of software even more limited, but also the formal notions for basins of attraction are often lacking. In this setting, the difficulty both for theory and computation arises from the fact that states may be ele- ments of several distinct basins. In this paper we address this topic by partitioning the state space into sets that are committed to the same attractors. These commitment sets can easily be generalised to sets that are equivalent w.r.t. the long-term behaviours of pre-selected nodes which leads us to the notions of markers and phenotypes which we illustrate in a case study on bladder tumorigenesis. For every concept we propose equivalent CTL model checking queries and an extension of the state of the art model checking software NuSMV is made available that is capa- ble of computing the respective sets. All notions are fully integrated as three new modules in our Python package PyBoolNet, including functions for visualising the basins, commitment sets and phenotypes as quotient graphs and pie charts

Boolean analysis of lateral inhibition

We study Boolean networks which are simple spatial models of the highly conserved Delta–Notch system. The models assume the inhibition of Delta in each cell by Notch in the same cell, and the activation of Notch in presence of Delta in surrounding cells. We consider fully asynchronous dynamics over undirected graphs representing the neighbour relation between cells. In this framework, one can show that all attractors are fixed points for the system, independently of the neighbour relation, for instance by using known properties of simplified versions of the models, where only one species per cell is defined. The fixed points correspond to the so-called fine-grained “patterns” that emerge in discrete and continuous modelling of lateral inhibition. We study the reachability of fixed points, giving a characterisation of the trap spaces and the basins of attraction for both the full and the simplified models. In addition, we use a characterisation of the trap spaces to investigate the robustness of patterns to perturbations. The results of this qualitative analysis can complement and guide simulation-based approaches, and serve as a basis for the investigation of more complex mechanisms

Approximating attractors of Boolean networks by iterative CTL model checking

This paper introduces the notion of approximating asynchronous attractors of Boolean networks by minimal trap spaces. We define three criteria for determining the quality of an approximation: “faithfulness” which requires that the oscillating variables of all attractors in a trap space correspond to their dimensions, “univocality” which requires that there is a unique attractor in each trap space, and “completeness” which requires that there are no attractors outside of a given set of trap spaces. Each is a reachability property for which we give equivalent model checking queries. Whereas faithfulness and univocality can be decided by model checking the corresponding subnetworks, the naive query for completeness must be evaluated on the full state space. Our main result is an alternative approach which is based on the iterative refinement of an initially poor approximation. The algorithm detects so-called autonomous sets in the interaction graph, variables that contain all their regulators, and considers their intersection and extension in order to perform model checking on the smallest possible state spaces. A benchmark, in which we apply the algorithm to 18 published Boolean networks, is given. In each case, the minimal trap spaces are faithful, univocal, and complete, which suggests that they are in general good approximations for the asymptotics of Boolean networks

Unraveling the regulation of mTORC2 using logical modeling

Background The mammalian target of rapamycin (mTOR) is a regulator of cell proliferation, cell growth and apoptosis working through two distinct complexes: mTORC1 and mTORC2. Although much is known about the activation and inactivation of mTORC1, the processes controlling mTORC2 remain poorly characterized. Experimental and modeling studies have attempted to explain the regulation of mTORC2 but have yielded several conflicting hypotheses. More specifically, the Phosphoinositide 3-kinase (PI3K) pathway was shown to be involved in this process, but the identity of the kinase interacting with and regulating mTORC2 remains to be determined (Cybulski and Hall, Trends Biochem Sci 34:620-7, 2009). Method We performed a literature search and identified 5 published hypotheses describing mTORC2 regulation. Based on these hypotheses, we built logical models, not only for each single hypothesis but also for all combinations and possible mechanisms among them. Based on data provided by the original studies, a systematic analysis of all models was performed. Results We were able to find models that account for experimental observations from every original study, but do not require all 5 hypotheses to be implemented. Surprisingly, all hypotheses were in agreement with all tested data gathered from the different studies and PI3K was identified as an essential regulator of mTORC2. Conclusion The results and additional data suggest that more than one regulator is necessary to explain the behavior of mTORC2. Finally, this study proposes a new experiment to validate mTORC1 as second essential regulator