The control of biological systems presents interesting applications such as cell reprogramming or drug target identification. A common type of control strategy consists in a set of interventions that, by fixing the values of some variables, force the system to evolve to a desired state. This work presents a new approach for finding control strategies in biological systems modeled by Boolean networks. In this context, we explore the properties of trap spaces, subspaces of the state space which the dynamics cannot leave. Trap spaces for biological networks can often be efficiently computed, and provide useful approximations of attraction basins. Our approach provides control strategies for a target phenotype that are based on interventions that allow the control to be eventually released. Moreover, our method can incorporate information about the attractors to find new control strategies that would escape usual percolation-based methods. We show the applicability of our approach to two cell fate decision models.

Temporal networks are a powerful tool for studying the dynamic nature of a wide range of real-world complex systems, including social, biological and physical systems. In particular, detection of dynamic communities within these networks can help identify important cohesive structures and fundamental mechanisms driving systems behaviour. However, when working with real-world systems, available data is often limited and sparse, due to missing data on systems entities, their evolution and interactions, as well as uncertainty regarding temporal resolution. This can hinder accurate representation of the system over time and result in incomplete or biased community dynamics. In this paper, we consider established methods for community detection and, using synthetic data experiments and real-world case studies, we evaluate the impact of data sparsity on the quality of identified dynamic communities. Our results give valuable insights on the evolution of systems with sparse data, which are less studied in existing literature, but are frequently encountered in real-world applications.

SummaryTuring patterns1are well-known self-organising systems that can form spots, stripes, or labyrinths. They represent a major theory of patterning in tissue organisation, due to their remarkable similarity to some natural patterns, such as skin pigmentation in zebrafish2, digit spacing3,4, and many others. The involvement of Turing patterns in biology has been debated because of their stringent fine-tuning requirements, where patterns only occur within a small subset of parameters5,6. This has complicated the engineering of a synthetic gene circuit for Turing patterns from first principles, even though natural genetic Turing networks have been successfully identified4,7. Here, we engineered a synthetic genetic reaction-diffusion system where three nodes interact according to a non-classical Turing network with improved parametric robustness6. The system was optimised inE. coliand reproducibly generated stationary, periodic, concentric stripe patterns in growing colonies. The patterns were successfully reproduced with a partial differential equation model, in a parameter regime obtained by fitting to experimental data. Our synthetic Turing system can contribute to novel nanotechnologies, such as patterned biomaterial deposition8,9, and provide insights into developmental patterning programs10.

Boolean models provide an intuitive framework for the investigation of complex biological networks. Dynamics that implement asynchronous update rules, in particular, can help embody the complexity arising from non-deterministic behavior. These transition systems allow for the emergence of complex attractors, cyclic subgraphs that capture oscillating asymptotic behavior. Techniques that explore and attempt to describe the structures of these attractors have received limited attention. In this context, the incorporation of process rate information may yield additional insights into dynamical patterns. Here, we propose to use a spectral clustering algorithm on the kinetic rate matrix of time-continuous Boolean networks to uncover dynamic structures within cyclic attractors. The Robust Perron Cluster Analysis (PCCA+) can be used to unravel metastable sets in Markov jump processes, i.e. sets in which a system remains for a long time before it switches to another metastable set. As a proof-of-concept, we apply this method to Boolean models of the mammalian cell cycle. By considering the categorization of transitions as either slow or fast, we investigate the impact of time information on the emergence of significant sub-structures.

Turing patterns are self-organizing systems that can form spots, stripes, or labyrinths. Proposed examples in tissue organization include zebrafish pigmentation, digit spacing, and many others. The theory of Turing patterns in biology has been debated because of their stringent fine-tuning requirements, where patterns only occur within a small subset of parameters. This has complicated the engineering of synthetic Turing gene circuits from first principles, although natural genetic Turing networks have been identified. Here, we engineered a synthetic genetic reaction-diffusion system where three nodes interact according to a non-classical Turing network with improved parametric robustness. The system reproducibly generated stationary, periodic, concentric stripe patterns in growing E. coli colonies. A partial differential equation model reproduced the patterns, with a Turing parameter regime obtained by fitting to experimental data. Our synthetic Turing system can contribute to nanotechnologies, such as patterned biomaterial deposition, and provide insights into developmental patterning programs. A record of this paper's transparent peer review process is included in the supplemental information.

Bioregionalization methods allow us to classify and map biogeographic units using data on species composition and traits. Here, we reviewed the evolution of the field during the last 70 years, seeking to summarize its history, and identify gaps and future avenues for research. Our results show that the aim of the studies using bioregionalization methods changed in time. First, bioregionalization were used to unveil the drivers of the observed spatial patterns of biodiversity on Earth, and to understand the role of dispersal limitations on the evolutionary history of clades, but recently, these methods are mostly used for conservation management. Further, data used to map biodiversity regions, the ones that we are now defining conservation strategies, are taxonomically and geographically biased, with a large percentage of the papers using vertebrate data from developed continents/countries. Finally, we show how key papers in the field, the ones with most citations, heavily depend on expert criteria and non-reproducible workflows, preventing direct comparison of maps of bioregions from different papers. Following our findings, we identified 3 gaps for the advance in the field, 1) We need to move beyond maps of vertebrate composition. Ideally, we need to increase the taxonomic diversity of the studies, but also to add other type of information, like data on species traits, genetic diversity, or phylogenetic distances. 2) we need reproducible and standardized methods 3) we need to further explore the temporal dimension of bioregions, to understand how they evolved through time.

The study of control mechanisms of biological systems allows for interesting applications in bioengineering and medicine, for instance in cell reprogramming or drug target identification. A control strategy often consists of a set of interventions that, by fixing the values of some components, ensure that the long term dynamics of the controlled system is in a desired state. A common approach to control in the Boolean framework consists in checking how the fixed values propagate through the network, to establish whether the effect of percolating the interventions is sufficient to induce the target state. Although methods based uniquely on value percolation allow for efficient computation, they can miss many control strategies. Exhaustive methods for control strategy identification, on the other hand, often entail high computational costs. In order to increase the number of control strategies identified while still benefiting from an efficient implementation, we introduce the use of trap spaces, subspaces of the state space that are closed with respect to the dynamics, and that can usually be easily computed in biological networks. This work presents a method based on value percolation that uses trap spaces to uncover new control strategies. It allows for node interventions, which fix the value of certain components, and edge interventions, which fix the effect that one component has on another. The method is implemented using Answer Set Programming, extending an existing efficient implementation of value percolation to allow for the use of trap spaces and edge control. The applicability of the approach is studied for different control targets in a biological case study, identifying in all cases new control strategies. The method presented here provides a new tool for control strategy identification in Boolean networks that allows for more diversity of interventions and for the possibility of efficiently finding new control strategies that would escape usual percolation-based methods, widening the possibility for potential applications

Understanding control mechanisms in biological systems plays a crucial role in important applications, for instance in cell reprogramming. Boolean modeling allows the identification of possible efficient strategies, helping to reduce the usually high and time-consuming experimental efforts. Available approaches to control strategy identification usually focus either on attractor or phenotype control, and are unable to deal with more complex control problems, for instance phenotype avoidance. They also fail to capture, in many situations, all possible minimal strategies, finding instead only sub-optimal solutions. In order to fill these gaps, we present a novel approach to control strategy identification in Boolean networks based on model checking. The method is guaranteed to identify all minimal control strategies, and provides maximal flexibility in the definition of the control target. We investigate the applicability of the approach by considering a range of control problems for different biological systems, comparing the results, where possible, to those obtained by alternative control methods.

Control of Boolean networks enables important medical and biological applications. At the core of many approaches is value percolation, by virtue of its simplicity and ease of implementation. Methods based uniquely on percolation can however miss many control strategies. We previously introduced a new method which, using the network's trap spaces, can uncover additional sets of interventions. In this work we present a highly efficient implementation of this methodology based on Answer Set Programming, allowing for simple and fast application to biological networks, as illustrated with some cases studies of cell reprogramming.

One of the main theoretical questions in the field of discrete regulatory networks is the question what aspects of the dynamics - the structure of the state transition graph - are already imposed by structural descriptions of the network such as the interaction graph. For Boolean networks, prior work has concentrated on different versions of the Thomas conjectures that link feedback cycles in the network structure to attractor properties. Other approaches check algorithmically whether certain properties hold true for all models sharing specific structural constraints, e.g. by using model checking techniques. In this work we investigate the behavior of the pool of Boolean networks in agreement with a given interaction graph using a different approach. Grouping together states that are updated consistently across the pool we derive an equivalence relation and analyze a corresponding quotient graph on the state space. By construction this graph yields information about the dynamics of all functions in the pool. Our main result is that this graph can be computed efficiently without enumerating and analyzing all individual functions. This opens up new possibilities for applications, where such model pools arise when modeling under uncertainty.

Motivation: Capturing the molecular diversity of living cells is not straightforward. One approach is to measure molecular markers that serve as indicators of specific biological conditions or phenotypes. This is particularly relevant in modern medicine to provide precise diagnostics and pinpoint the best treatment for each patient. The challenge is to select a minimal set of markers whose activity patterns are in correspondence with the phenotypes of interest. Results: This article approaches the marker detection problem in the context of discrete phenotypes which arise, for example, from Boolean models of cellular networks. Mathematically this poses a combinatorial optimization problem with many answers. We propose a solution to this optimization problem that is based on the modelling language answer set programming (ASP). A case study of a death cell receptor network illustrates the methodology. Discussion and code: For code, discussions and reporting errors visit https://github.com/hklarner/detection_of_markers_for_discrete_phenotypes.

Boolean delay equations (BDEs), with their relatively simple and intuitive mode of modelling, have been used in many research areas including, for example, climate dynamics and earthquake propagation. Their application to biological systems has been scarce and limited to the molecular level. Here, we derive and present two BDE models. One is directly derived from a previously published ordinary differential equation (ODE) model for the bovine estrous cycle, whereas the second model includes a modification of a particular biological mechanism. We not only compare the simulation results from the BDE models with the trajectories of the ODE model, but also validate the BDE models with two additional numerical experiments. One experiment induces a switch in the oscillatory pattern upon changes in the model parameters, and the other simulates the administration of a hormone that is known to shift the estrous cycle in time. The models presented here are the first BDE models for hormonal oscillators, and the first BDE models for drug administration. Even though automatic parameter estimation still remains challenging, our results support the role of BDEs as a framework for the systematic modelling of complex biological oscillators.

The formation of spatial structures lies at the heart of developmental processes. However, many of the underlying gene regulatory and biochemical processes remain poorly understood. Turing patterns constitute a main candidate to explain such processes, but they appear sensitive to fluctuations and variations in kinetic parameters, raising the question of how they may be adopted and realised in naturally evolved systems. The vast majority of mathematical studies of Turing patterns have used continuous models specified in terms of partial differential equations. Here, we complement this work by studying Turing patterns using discrete cellular automata models. We perform a large-scale study on all possible two-species networks and find the same Turing pattern producing networks as in the continuous framework. In contrast to continuous models, however, we find these Turing pattern topologies to be substantially more robust to changes in the parameters of the model. We also find that diffusion-driven instabilities are substantially weaker predictors for Turing patterns in our discrete modelling framework in comparison to the continuous case, in the sense that the presence of an instability does not guarantee a pattern emerging in simulations. We show that a more refined criterion constitutes a stronger predictor. The similarity of the results for the two modelling frameworks suggests a deeper underlying principle of Turing mechanisms in nature. Together with the larger robustness in the discrete case this suggests that Turing patterns may be more robust than previously thought.

Motivation Cell classifiers are synthetic bio-devices performing type-specific in vivo classification. The circuits identify a cell state based on its molecular fingerprint. In particular, the classifiers may be designed to recognize cancerous cells and trigger their apoptosis, shaping a novel therapy for cancer patients. Recently, we introduced a new theoretical design of such devices employing distributed classifiers. Here, a group of single-circuit classifiers decides collectively according to a pre-defined threshold function whether a cell is cancerous. The multi-circuit architecture has shown the potential to predict the cell condition with high accuracy. However, lack of far-reaching machinery to design and evaluate distributed cell classifiers, in particular, assessing their robustness to noise and novel information, makes their application limited. Results In this study, we present a comprehensive framework for designing and evaluating miRNA-based distributed cell classifiers comprising data simulation, pre-processing, and an extensive testing scheme. We develop optimization criteria that allow increasing the accuracy and robustness of classifiers to noise and novel information as shown in simulated and real-world data studies. The evaluation performed on cancer data demonstrates that distributed classifiers outperform single-circuit designs in terms of prediction accuracy. Our classifiers include relevant miRNAs previously described in the literature, as well as more complex regulation patterns included in the data. Availability The code and data are available at: https://github.com/MelaniaNowicka/RAccoon . Contact m.nowicka@fu-berlin.de

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.

We investigate how classifiers for Boolean networks (BNs) can be constructed and modified under constraints. A typical constraint is to observe only states in attractors or even more specifically steady states of BNs. Steady states of BNs are one of the most interesting features for application. Large models can possess many steady states. In the typical scenario motivating this paper we start from a Boolean model with a given classification of the state space into phenotypes defined by high-level readout components. In order to link molecular biomarkers with experimental design, we search for alternative components suitable for the given classification task. This is useful for modelers of regulatory networks for suggesting experiments and measurements based on their models. It can also help to explain causal relations between components and phenotypes. To tackle this problem we need to use the structure of the BN and the constraints. This calls for an algebraic approach. Indeed we demonstrate that this problem can be reformulated into the language of algebraic geometry. While already interesting in itself, this allows us to use Gröbner bases to construct an algorithm for finding such classifiers. We demonstrate the usefulness of this algorithm as a proof of concept on a model with 25 components.

Background Although several studies have provided insights into the role of long non-coding RNAs (lncRNAs), the majority of them have unknown function. Recent evidence has shown the importance of both lncRNAs and chromatin interactions in transcriptional regulation. Although network-based methods, mainly exploiting gene-lncRNA co-expression, have been applied to characterize lncRNA of unknown function by means of ’guilt-by-association’, no strategy exists so far which identifies mRNA-lncRNA functional modules based on the 3D chromatin interaction graph. Results To better understand the function of chromatin interactions in the context of lncRNA-mediated gene regulation, we have developed a multi-step graph analysis approach to examine the RNA polymerase II ChIA-PET chromatin interaction network in the K562 human cell line. We have annotated the network with gene and lncRNA coordinates, and chromatin states from the ENCODE project. We used centrality measures, as well as an adaptation of our previously developed Markov State Models (MSM) clustering method, to gain a better understanding of lncRNAs in transcriptional regulation. The novelty of our approach resides in the detection of fuzzy regulatory modules based on network properties and their optimization based on co-expression analysis between genes and gene-lncRNA pairs. This results in our method returning more bona fide regulatory modules than other state-of-the art approaches for clustering on graphs. Conclusions Interestingly, we find that lncRNA network hubs tend to be significantly enriched in evolutionary conserved lncRNAs and enhancer-like functions. We validated regulatory functions for well known lncRNAs, such as MALAT1 and the enhancer-like lncRNA FALEC. In addition, by investigating the modular structure of bigger components we mine putative regulatory functions for uncharacterized lncRNAs.

Cell classifiers are decision-making synthetic circuits that allow in vivo cell-type classification. Their design is based on finding a relationship between differential expression of miRNAs and the cell condition. Such biological devices have shown potential to become a valuable tool in cancer treatment as a new type-specific cell targeting approach. So far, only single-circuit classifiers were designed in this context. However, reliable designs come with high complexity, making them difficult to assemble in the lab. Here, we apply so-called Distributed Classifiers (DC) consisting of simple single circuits, that decide collectively according to a threshold function. Such architecture potentially simplifies the assembly process and provides design flexibility. We present a genetic algorithm that allows the design and optimization of DCs. Breast cancer case studies show that DCs perform with high accuracy on real-world data. Optimized classifiers capture biologically relevant miRNAs that are cancer-type specific. The comparison to a single-circuit classifier design approach shows that DCs perform with significantly higher accuracy than individual circuits. The algorithm is implemented as an open source tool.

Asymptotic behaviors are often of particular interest when analyzing Boolean networks that represent biological systems such as signal transduction or gene regulatory networks. Methods based on a generalization of the steady state notion, the so-called trap spaces, can be exploited to investigate attractor properties as well as for model reduction techniques. In this paper, we propose a novel optimization-based method for computing all minimal and maximal trap spaces and motivate their use. In particular, we add a new result yielding a lower bound for the number of cyclic attractors and illustrate the methods with a study of a MAPK pathway model. To test the efficiency and scalability of the method, we compare the performance of the ILP solver gurobi with the ASP solver potassco in a benchmark of random networks.

Systems biology studies the structure and dynamics of biological systems using mathematical approaches. Bottom-up approaches create models from prior knowledge but usually cannot cope with uncertainty, whereas top-down approaches infer models directly from data using statistical methods but mostly neglect valuable known information from former studies. Here, we want to present a workflow that includes prior knowledge while allowing for uncertainty in the modeling process. We build not one but all possible models that arise from the uncertainty using logical modeling and subsequently filter for those models in agreement with data in a top-down manner. This approach enables us to investigate new and more complex biological research questions, however, the encoding in such a framework is often not obvious and thus not easily accessible for researcher from life sciences. To mitigate this problem, we formulate a pipeline with specific templates to address some research questions common in signaling network analysis. To illustrate the potential of this approach, we applied the pipeline to growth factor signaling processes in two renal cancer cell lines. These two cell lines originate from similar tissue, but surprisingly showed a very different behavior toward the cancer drug Sorafenib. Thus our aim was to explore differences between these cell lines regarding three sources of uncertainty in one analysis: possible targets of Sorafenib, crosstalk between involved pathways, and the effect of a mutation in mammalian target of Rapamycin (mTOR) in one of the cell lines. We were able to show that the model pools from the cell lines are disjoint, thus the discrepancies in behavior originate from differences in the cellular wiring. Also the mutation in mTOR is not affecting its activity in the pathway. The results on Sorafenib, while not fully clarifying the mechanisms involved, illustrate the potential of this analysis for generating new hypotheses.