Structure and behavior in Boolean monotonic model pools.

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.

Modelling Oscillatory Patterns in the Bovine Estrous Cycle with Boolean Delay Equations.

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 design principles of discrete turing patterning systems.

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.

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.

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 modeling, e.g., in systems biology. Yet, there are currently very few tools available that are able to compute and visualize 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 elements 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 generalized to sets that are equivalent w.r.t. the long-term behaviors 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 capable of computing the respective sets. All notions are fully integrated as three new modules in our Python package PyBoolNet, including functions for visualizing the basins, commitment sets, and phenotypes as quotient graphs and pie charts.

Identifying lncRNA-mediated regulatory modules via ChIA-PET network analysis.

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.

Evaluating Uncertainty in Signaling Networks Using Logical Modeling.

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.

Designing miRNA-Based Synthetic Cell Classifier Circuits Using Answer Set Programming.

Cell classifier circuits are synthetic biological circuits capable of distinguishing between different cell states depending on specific cellular markers and engendering a state-specific response. An example are classifiers for cancer cells that recognize whether a cell is healthy or diseased based on its miRNA fingerprint and trigger cell apoptosis in the latter case. Binarization of continuous miRNA expression levels allows to formalize a classifier as a Boolean function whose output codes for the cell condition. In this framework, the classifier design problem consists of finding a Boolean function capable of reproducing correct labelings of miRNA profiles. The specifications of such a function can then be used as a blueprint for constructing a corresponding circuit in the lab. To find an optimal classifier both in terms of performance and reliability, however, accuracy, design simplicity and constraints derived from availability of molcular building blocks for the classifiers all need to be taken into account. These complexities translate to computational difficulties, so currently available methods explore only part of the design space and consequently are only capable of calculating locally optimal designs. We present a computational approach for finding globally optimal classifier circuits based on binarized miRNA datasets using Answer Set Programming for efficient scanning of the entire search space. Additionally, the method is capable of computing all optimal solutions, allowing for comparison between optimal classifier designs and identification of key features. Several case studies illustrate the applicability of the approach and highlight the quality of results in comparison with a state of the art method. The method is fully implemented and a comprehensive performance analysis demonstrates its reliability and scalability.

Relationship of acute axonal damage, Wallerian degeneration, and clinical disability in multiple sclerosis.

BACKGROUND: Axonal damage and loss substantially contribute to the incremental accumulation of clinical disability in progressive multiple sclerosis. Here, we assessed the amount of Wallerian degeneration in brain tissue of multiple sclerosis patients in relation to demyelinating lesion activity and asked whether a transient blockade of Wallerian degeneration decreases axonal loss and clinical disability in a mouse model of inflammatory demyelination. METHODS: Wallerian degeneration and acute axonal damage were determined immunohistochemically in the periplaque white matter of multiple sclerosis patients with early actively demyelinating lesions, chronic active lesions, and inactive lesions. Furthermore, we studied the effects of Wallerian degeneration blockage on clinical severity, inflammatory pathology, acute axonal damage, and long-term axonal loss in experimental autoimmune encephalomyelitis using Wallerian degeneration slow (Wld S ) mutant mice. RESULTS: The highest numbers of axons undergoing Wallerian degeneration were found in the perilesional white matter of multiple sclerosis patients early in the disease course and with actively demyelinating lesions. Furthermore, Wallerian degeneration was more abundant in patients harboring chronic active as compared to chronic inactive lesions. No co-localization of neuropeptide Y-Y1 receptor, a bona fide immunohistochemical marker of Wallerian degeneration, with amyloid precursor protein, frequently used as an indicator of acute axonal transport disturbance, was observed in human and mouse tissue, indicating distinct axon-degenerative processes. Experimentally, a delay of Wallerian degeneration, as observed in Wld S mice, did not result in a reduction of clinical disability or acute axonal damage in experimental autoimmune encephalomyelitis, further supporting that acute axonal damage as reflected by axonal transport disturbances does not share common molecular mechanisms with Wallerian degeneration. Furthermore, delaying Wallerian degeneration did not result in a net rescue of axons in late lesion stages of experimental autoimmune encephalomyelitis. CONCLUSIONS: Our data indicate that in multiple sclerosis, ongoing demyelination in focal lesions is associated with axonal degeneration in the perilesional white matter, supporting a role for focal pathology in diffuse white matter damage. Also, our results suggest that interfering with Wallerian degeneration in inflammatory demyelination does not suffice to prevent acute axonal damage and finally axonal loss.

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.

PyBoolNet: a python package for the generation, analysis and visualization of boolean networks.

Motivation: The goal of this project is to provide a simple interface to working with Boolean networks. Emphasis is put on easy access to a large number of common tasks including the generation and manipulation of networks, attractor and basin computation, model checking and trap space computation, execution of established graph algorithms as well as graph drawing and layouts. Results: P y B ool N et is a Python package for working with Boolean networks that supports simple access to model checking via N u SMV, standard graph algorithms via N etwork X and visualization via dot . In addition, state of the art attractor computation exploiting P otassco ASP is implemented. The package is function-based and uses only native Python and N etwork X data types. Availability and Implementation: https://github.com/hklarner/PyBoolNet. Contact: hannes.klarner@fu-berlin.de.

Data-driven optimizations for model checking of multi-valued regulatory networks.

The model checking method has been long since established as an important tool for modelling and reverse engineering of biological systems. However, due to a high complexity of both the method and the biological systems, this approach often requires a vast amount of computational resources. In this article we show that by reducing the expressivity of the method we can gain performance while still being able to use all biologically relevant data. We utilize this approach to conduct a study of mutations in the EGFR signalling, motivated by a paper from Klinger et al. (2013). Here we aim at constructing approximated models of multiple cell-lines from sizeable sets of experimental data. Due to cancerous mutations in each cell line, there is a high degree of parameter uncertainty and the study would not be practically tractable without the performance optimizations described here.

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.

Complementing ODE-Based System Analysis Using Boolean Networks Derived from an Euler-Like Transformation.

In this paper, we present a systematic transition scheme for a large class of ordinary differential equations (ODEs) into Boolean networks. Our transition scheme can be applied to any system of ODEs whose right hand sides can be written as sums and products of monotone functions. It performs an Euler-like step which uses the signs of the right hand sides to obtain the Boolean update functions for every variable of the corresponding discrete model. The discrete model can, on one hand, be considered as another representation of the biological system or, alternatively, it can be used to further the analysis of the original ODE model. Since the generic transformation method does not guarantee any property conservation, a subsequent validation step is required. Depending on the purpose of the model this step can be based on experimental data or ODE simulations and characteristics. Analysis of the resulting Boolean model, both on its own and in comparison with the ODE model, then allows to investigate system properties not accessible in a purely continuous setting. The method is exemplarily applied to a previously published model of the bovine estrous cycle, which leads to new insights regarding the regulation among the components, and also indicates strongly that the system is tailored to generate stable oscillations.

Logical-continuous modelling of post-translationally regulated bistability of curli fiber expression in Escherichia coli.

BACKGROUND: Bacteria have developed a repertoire of signalling mechanisms that enable adaptive responses to fluctuating environmental conditions. The formation of biofilm, for example, allows persisting in times of external stresses, e.g. induced by antibiotics or a lack of nutrients. Adhesive curli fibers, the major extracellular matrix components in Escherichia coli biofilms, exhibit heterogeneous expression in isogenic cells exposed to identical external conditions. The dynamical mechanisms underlying this heterogeneity remain poorly understood. In this work, we elucidate the potential role of post-translational bistability as a source for this heterogeneity. RESULTS: We introduce a structured modelling workflow combining logical network topology analysis with time-continuous deterministic and stochastic modelling. The aim is to evaluate the topological structure of the underlying signalling network and to identify and analyse model parameterisations that satisfy observations from a set of genetic knockout experiments. Our work supports the hypothesis that the phenotypic heterogeneity of curli expression in biofilm cells is induced by bistable regulation at the post-translational level. Stochastic modelling suggests diverse noise-induced switching behaviours between the stable states, depending on the expression levels of the c-di-GMP-producing (diguanylate cyclases, DGCs) and -degrading (phosphodiesterases, PDEs) enzymes and reveals the quantitative difference in stable c-di-GMP levels between distinct phenotypes. The most dominant type of behaviour is characterised by a fast switching from curli-off to curli-on with a slow switching in the reverse direction and the second most dominant type is a long-term differentiation into curli-on or curli-off cells. This behaviour may implicate an intrinsic feature of the system allowing for a fast adaptive response (curli-on) versus a slow transition to the curli-off state, in line with experimental observations. CONCLUSION: The combination of logical and continuous modelling enables a thorough analysis of different determinants of bistable regulation, i.e. network topology and biochemical kinetics, and allows for an incorporation of experimental data from heterogeneous sources. Our approach yields a mechanistic explanation for the phenotypic heterogeneity of curli fiber expression. Furthermore, the presented work provides a detailed insight into the interactions between the multiple DGC- and PDE-type enzymes and the role of c-di-GMP in dynamical regulation of cellular decisions.

Preservation of dynamic properties in qualitative modeling frameworks for gene regulatory networks.

Mathematical modeling often helps to provide a systems perspective on gene regulatory networks. In particular, qualitative approaches are useful when detailed kinetic information is lacking. Multiple methods have been developed that implement qualitative information in different ways, e.g., in purely discrete or hybrid discrete/continuous models. In this paper, we compare the discrete asynchronous logical modeling formalism for gene regulatory networks due to R. Thomas with piecewise affine differential equation models. We provide a local characterization of the qualitative dynamics of a piecewise affine differential equation model using the discrete dynamics of a corresponding Thomas model. Based on this result, we investigate the consistency of higher-level dynamical properties such as attractor characteristics and reachability. We show that although the two approaches are based on equivalent information, the resulting qualitative dynamics are different. In particular, the dynamics of the piecewise affine differential equation model is not a simple refinement of the dynamics of the Thomas model.

Analysis and characterization of asynchronous state transition graphs using extremal states.

Logical modeling of biological regulatory networks gives rise to a representation of the system's dynamics as a so-called state transition graph. Analysis of such a graph in its entirety allows for a comprehensive understanding of the functionalities and behavior of the modeled system. However, the size of the vertex set of the graph is exponential in the number of the network components making analysis costly, motivating development of reduction methods. In this paper, we present results allowing for a complete description of an asynchronous state transition graph of a Thomas network solely based on the analysis of the subgraph induced by certain extremal states. Utilizing this notion, we compare the behavior of a simple multivalued network and a corresponding Boolean network and analyze the conservation of dynamical properties between them. Understanding the relation between such coarser and finer models is a necessary step toward meaningful network reduction as well as model refinement methods.

Time series dependent analysis of unparametrized Thomas networks.

This paper is concerned with the analysis of labeled Thomas networks using discrete time series. It focuses on refining the given edge labels and on assessing the data quality. The results are aimed at being exploitable for experimental design and include the prediction of new activatory or inhibitory effects of given interactions and yet unobserved oscillations of specific components in between specific sampling intervals. On the formal side, we generalize the concept of edge labels and introduce a discrete time series interpretation. This interpretation features two original concepts: 1) Incomplete measurements are admissible, and 2) it allows qualitative assumptions about the changes in gene expression by means of monotonicity. On the computational side, we provide a Python script, erda.py, that automates the suggested workflow by model checking and constraint satisfaction. We illustrate the workflow by investigating the yeast network IRMA.

Analysis of discrete bioregulatory networks using symbolic steady states.

A discrete model of a biological regulatory network can be represented by a discrete function that contains all available information on interactions between network components and the rules governing the evolution of the network in a finite state space. Since the state space size grows exponentially with the number of network components, analysis of large networks is a complex problem. In this paper, we introduce the notion of symbolic steady state that allows us to identify subnetworks that govern the dynamics of the original network in some region of state space. We state rules to explicitly construct attractors of the system from subnetwork attractors. Using the results, we formulate sufficient conditions for the existence of multiple attractors resp. a cyclic attractor based on the existence of positive resp. negative feedback circuits in the graph representing the structure of the system. In addition, we discuss approaches to finding symbolic steady states. We focus both on dynamics derived via synchronous as well as asynchronous update rules. Lastly, we illustrate the results by analyzing a model of T helper cell differentiation.

Over-expression of alpha-synuclein in the nervous system enhances axonal degeneration after peripheral nerve lesion in a transgenic mouse strain.

Wallerian degeneration in peripheral nerves occurs after a traumatic insult when the distal nerve part degenerates while peripheral macrophages enter the nerve stump and remove the accruing debris by phagozytosis. We used an experimental model to investigate the effect of either the absence or over-expression of alpha-synuclein (alpha-syn) after transecting the sciatic nerves of mice. alpha-Synuclein is a major component of Lewy bodies and its aggregation results in a premature destruction of nerve cells. It has also been found present in different peripheral nerves but its role in the axon remains still unclear. Following sciatic nerve transection in different mouse strains, we investigated the numbers of invading macrophages, the amounts of remaining myelin and axons 6 days after injury. All mice showed clear signs of Wallerian degeneration, but transgenic mice expressing human wild-type alpha-syn showed lower numbers of invading macrophages, less preserved myelin and significantly lower numbers of preserved axons in comparison with either knockout mice or a mouse strain with a spontaneous deletion of alpha-syn. The use of protein aggregation filtration blots and paraffin-embedded tissue blots displayed depositions of alpha-syn aggregates within sciatic nerve axons of transgenic mice. Thicker myelin sheaths and higher numbers of mitochondria were detected in old alpha-syn transgenic mice. In a human sural nerve, alpha-syn could also be identified within axons. Thus, alpha-syn and its aggregates are not only a component of Lewy bodies and synapses but also of axons and these aggregates might interfere with axonal transport. alpha-Synuclein transgenic mice represent an appropriate model for investigations on axonal transport in neurodegenerative diseases.