Logical modelling of Drosophila signalling pathways

A meta-analysis of Boolean network models reveals design principles of gene regulatory networks

Gene regulatory networks (GRNs) play a central role in cellular decision-making. Understanding their structure and how it impacts their dynamics constitutes thus a fundamental biological question. GRNs are frequently modeled as Boolean networks, which are intuitive, simple to describe, and can yield qualitative results even when data are sparse. We assembled the largest repository of expert-curated Boolean GRN models. A meta-analysis of this diverse set of models reveals several design principles. GRNs exhibit more canalization, redundancy, and stable dynamics than expected. Moreover, they are enriched for certain recurring network motifs. This raises the important question why evolution favors these design mechanisms.

Models of Cell Processes are Far from the Edge of Chaos

Complex living systems are thought to exist at the "edge of chaos" separating the ordered dynamics of robust function from the disordered dynamics of rapid environmental adaptation. Here, a deeper inspection of 72 experimentally supported discrete dynamical models of cell processes reveals previously unobserved order on long time scales, suggesting greater rigidity in these systems than was previously conjectured. We find that propagation of internal perturbations is transient in most cases, and that even when large perturbation cascades persist, their phenotypic effects are often minimal. Moreover, we find evidence that stochasticity and desynchronization can lead to increased recovery from regulatory perturbation cascades. Our analysis relies on new measures that quantify the tendency of perturbations to spread through a discrete dynamical system. Computing these measures was not feasible using current methodology; thus, we developed a multipurpose CUDA-based simulation tool, which we have made available as the open-source Python library cubewalkers. Based on novel measures and simulations, our results suggest that-contrary to current theory-cell processes are ordered and far from the edge of chaos.

Reducing Boolean networks with backward equivalence

BACKGROUND

Boolean Networks (BNs) are a popular dynamical model in biology where the state of each component is represented by a variable taking binary values that express, for instance, activation/deactivation or high/low concentrations. Unfortunately, these models suffer from the state space explosion, i.e., there are exponentially many states in the number of BN variables, which hampers their analysis.

RESULTS

We present Boolean Backward Equivalence (BBE), a novel reduction technique for BNs which collapses system variables that, if initialized with same value, maintain matching values in all states. A large-scale validation on 86 models from two online model repositories reveals that BBE is effective, since it is able to reduce more than 90% of the models. Furthermore, on such models we also show that BBE brings notable analysis speed-ups, both in terms of state space generation and steady-state analysis. In several cases, BBE allowed the analysis of models that were originally intractable due to the complexity. On two selected case studies, we show how one can tune the reduction power of BBE using model-specific information to preserve all dynamics of interest, and selectively exclude behavior that does not have biological relevance.

CONCLUSIONS

BBE complements existing reduction methods, preserving properties that other reduction methods fail to reproduce, and vice versa. BBE drops all and only the dynamics, including attractors, originating from states where BBE-equivalent variables have been initialized with different activation values The remaining part of the dynamics is preserved exactly, including the length of the preserved attractors, and their reachability from given initial conditions, without adding any spurious behaviours. Given that BBE is a model-to-model reduction technique, it can be combined with further reduction methods for BNs.

Long-Term Memory Formation in Drosophila Depends on the 3'UTR of CPEB Gene orb2

Activation of local translation in neurites in response to stimulation is an important step in the formation of long-term memory (LTM). CPEB proteins are a family of translation factors involved in LTM formation. The Drosophila CPEB protein Orb2 plays an important role in the development and function of the nervous system. Mutations of the coding region of the orb2 gene have previously been shown to impair LTM formation. We found that a deletion of the 3'UTR of the orb2 gene similarly results in loss of LTM in Drosophila. As a result of the deletion, the content of the Orb2 protein remained the same in the neuron soma, but significantly decreased in synapses. Using RNA immunoprecipitation followed by high-throughput sequencing, we detected more than 6000 potential Orb2 mRNA targets expressed in the Drosophila brain. Importantly, deletion of the 3'UTR of orb2 mRNA also affected the localization of the Csp, Pyd, and Eya proteins, which are encoded by putative mRNA targets of Orb2. Therefore, the 3'UTR of the orb2 mRNA is important for the proper localization of Orb2 and other proteins in synapses of neurons and the brain as a whole, providing a molecular basis for LTM formation.

The basis of easy controllability in Boolean networks

Effective control of biological systems can often be achieved through the control of a surprisingly small number of distinct variables. We bring clarity to such results using the formalism of Boolean dynamical networks, analyzing the effectiveness of external control in selecting a desired final state when that state is among the original attractors of the dynamics. Analyzing 49 existing biological network models, we find strong numerical evidence that the average number of nodes that must be forced scales logarithmically with the number of original attractors. This suggests that biological networks may be typically easy to control even when the number of interacting components is large. We provide a theoretical explanation of the scaling by separating controlling nodes into three types: those that act as inputs, those that distinguish among attractors, and any remaining nodes. We further identify characteristics of dynamics that can invalidate this scaling, and speculate about how this relates more broadly to non-biological systems.

Boolean networks allow a simplified representation of interactions. Here, the authors systematically analyze regulation in dozens of biological Boolean networks, finding mathematical regularities that suggest biological systems could be controlled through a relatively small number of components.

Symbolic Coloured SCC Decomposition

Problems arising in many scientific disciplines are often modelled using edge-coloured directed graphs. These can be enormous in the number of both vertices and colours. Given such a graph, the original problem frequently translates to the detection of the graph’s strongly connected components, which is challenging at this scale.

We propose a new, symbolic algorithm that computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(p\cdot n\cdot \log n)$$\end{document}O(p·n·logn) symbolic steps, where p is the number of colours and n the number of vertices. We evaluate the algorithm using an experimental implementation based on Binary Decision Diagrams (BDDs) and large (up to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{48}$$\end{document}248) coloured graphs produced by models appearing in systems biology.

Topology-driven protein-protein interaction network analysis detects genetic sub-networks regulating reproductive capacity

Understanding the genetic regulation of organ structure is a fundamental problem in developmental biology. Here, we use egg-producing structures of insect ovaries, called ovarioles, to deduce systems-level gene regulatory relationships from quantitative functional genetic analysis. We previously showed that Hippo signalling, a conserved regulator of animal organ size, regulates ovariole number in Drosophila melanogaster. To comprehensively determine how Hippo signalling interacts with other pathways in this regulation, we screened all known signalling pathway genes, and identified Hpo-dependent and Hpo-independent signalling requirements. Network analysis of known protein-protein interactions among screen results identified independent gene regulatory sub-networks regulating one or both of ovariole number and egg laying. These sub-networks predict involvement of previously uncharacterised genes with higher accuracy than the original candidate screen. This shows that network analysis combining functional genetic and large-scale interaction data can predict function of novel genes regulating development.

Antifragility Predicts the Robustness and Evolvability of Biological Networks through Multi-Class Classification with a Convolutional Neural Network

Robustness and evolvability are essential properties to the evolution of biological networks. To determine if a biological network is robust and/or evolvable, it is required to compare its functions before and after mutations. However, this sometimes takes a high computational cost as the network size grows. Here, we develop a predictive method to estimate the robustness and evolvability of biological networks without an explicit comparison of functions. We measure antifragility in Boolean network models of biological systems and use this as the predictor. Antifragility occurs when a system benefits from external perturbations. By means of the differences of antifragility between the original and mutated biological networks, we train a convolutional neural network (CNN) and test it to classify the properties of robustness and evolvability. We found that our CNN model successfully classified the properties. Thus, we conclude that our antifragility measure can be used as a predictor of the robustness and evolvability of biological networks.

Conceptual and computational framework for logical modelling of biological networks deregulated in diseases

Mathematical models can serve as a tool to formalize biological knowledge from diverse sources, to investigate biological questions in a formal way, to test experimental hypotheses, to predict the effect of perturbations and to identify underlying mechanisms. We present a pipeline of computational tools that performs a series of analyses to explore a logical model's properties. A logical model of initiation of the metastatic process in cancer is used as a transversal example. We start by analysing the structure of the interaction network constructed from the literature or existing databases. Next, we show how to translate this network into a mathematical object, specifically a logical model, and how robustness analyses can be applied to it. We explore the visualization of the stable states, defined as specific attractors of the model, and match them to cellular fates or biological read-outs. With the different tools we present here, we explain how to assign to each solution of the model a probability and how to identify genetic interactions using mutant phenotype probabilities. Finally, we connect the model to relevant experimental data: we present how some data analyses can direct the construction of the network, and how the solutions of a mathematical model can also be compared with experimental data, with a particular focus on high-throughput data in cancer biology. A step-by-step tutorial is provided as a Supplementary Material and all models, tools and scripts are provided on an accompanying website: https://github.com/sysbio-curie/Logical_modelling_pipeline.

Parameter estimation of qualitative biological regulatory networks on high performance computing hardware

BACKGROUND

Biological Regulatory Networks (BRNs) are responsible for developmental and maintenance related functions in organisms. These functions are implemented by the dynamics of BRNs and are sensitive to regulations enforced by specific activators and inhibitors. The logical modeling formalism by René Thomas incorporates this sensitivity with a set of logical parameters modulated by available regulators, varying with time. With the increase in complexity of BRNs in terms of number of entities and their interactions, the task of parameters estimation becomes computationally expensive with existing sequential SMBioNET tool. We extend the existing sequential implementation of SMBioNET by using a data decomposition approach using a Java messaging library called MPJ Express. The approach divides the parameters space into different regions and each region is then explored in parallel on High Performance Computing (HPC) hardware.

RESULTS

The performance of the parallel approach is evaluated on BRNs of different sizes, and experimental results on multicore and cluster computers showed almost linear speed-up. This parallel code can be executed on a wide range of concurrent hardware including laptops equipped with multicore processors, and specialized distributed memory computer systems. To demonstrate the application of parallel implementation, we selected a case study of Hexosamine Biosynthetic Pathway (HBP) in cancer progression to identify potential therapeutic targets against cancer. A set of logical parameters were computed for HBP model that directs the biological system to a state of recovery. Furthermore, the parameters also suggest a potential therapeutic intervention that restores homeostasis. Additionally, the performance of parallel application was also evaluated on a network (comprising of 23 entities) of Fibroblast Growth Factor Signalling in Drosophila melanogaster.

CONCLUSIONS

Qualitative modeling framework is widely used for investigating dynamics of biological regulatory networks. However, computation of model parameters in qualitative modeling is computationally intensive. In this work, we presented results of our Java based parallel implementation that provides almost linear speed-up on both multicore and cluster platforms. The parallel implementation is available at https://psmbionet.github.io .

ASP-based method for the enumeration of attractors in non-deterministic synchronous and asynchronous multi-valued networks

BACKGROUND

This paper addresses the problem of finding attractors in biological regulatory networks. We focus here on non-deterministic synchronous and asynchronous multi-valued networks, modeled using automata networks (AN). AN is a general and well-suited formalism to study complex interactions between different components (genes, proteins,...). An attractor is a minimal trap domain, that is, a part of the state-transition graph that cannot be escaped. Such structures are terminal components of the dynamics and take the form of steady states (singleton) or complex compositions of cycles (non-singleton). Studying the effect of a disease or a mutation on an organism requires finding the attractors in the model to understand the long-term behaviors.

RESULTS

We present a computational logical method based on answer set programming (ASP) to identify all attractors. Performed without any network reduction, the method can be applied on any dynamical semantics. In this paper, we present the two most widespread non-deterministic semantics: the asynchronous and the synchronous updating modes. The logical approach goes through a complete enumeration of the states of the network in order to find the attractors without the necessity to construct the whole state-transition graph. We realize extensive computational experiments which show good performance and fit the expected theoretical results in the literature.

CONCLUSION

The originality of our approach lies on the exhaustive enumeration of all possible (sets of) states verifying the properties of an attractor thanks to the use of ASP. Our method is applied to non-deterministic semantics in two different schemes (asynchronous and synchronous). The merits of our methods are illustrated by applying them to biological examples of various sizes and comparing the results with some existing approaches. It turns out that our approach succeeds to exhaustively enumerate on a desktop computer, in a large model (100 components), all existing attractors up to a given size (20 states). This size is only limited by memory and computation time.

Qualitative Dynamical Modelling Can Formally Explain Mesoderm Specification and Predict Novel Developmental Phenotypes

Given the complexity of developmental networks, it is often difficult to predict the effect of genetic perturbations, even within coding genes. Regulatory factors generally have pleiotropic effects, exhibit partially redundant roles, and regulate highly interconnected pathways with ample cross-talk. Here, we delineate a logical model encompassing 48 components and 82 regulatory interactions involved in mesoderm specification during Drosophila development, thereby providing a formal integration of all available genetic information from the literature. The four main tissues derived from mesoderm correspond to alternative stable states. We demonstrate that the model can predict known mutant phenotypes and use it to systematically predict the effects of over 300 new, often non-intuitive, loss- and gain-of-function mutations, and combinations thereof. We further validated several novel predictions experimentally, thereby demonstrating the robustness of model. Logical modelling can thus contribute to formally explain and predict regulatory outcomes underlying cell fate decisions.

Designing Experiments to Discriminate Families of Logic Models

Logic models of signaling pathways are a promising way of building effective in silico functional models of a cell, in particular of signaling pathways. The automated learning of Boolean logic models describing signaling pathways can be achieved by training to phosphoproteomics data, which is particularly useful if it is measured upon different combinations of perturbations in a high-throughput fashion. However, in practice, the number and type of allowed perturbations are not exhaustive. Moreover, experimental data are unavoidably subjected to noise. As a result, the learning process results in a family of feasible logical networks rather than in a single model. This family is composed of logic models implementing different internal wirings for the system and therefore the predictions of experiments from this family may present a significant level of variability, and hence uncertainty. In this paper, we introduce a method based on Answer Set Programming to propose an optimal experimental design that aims to narrow down the variability (in terms of input-output behaviors) within families of logical models learned from experimental data. We study how the fitness with respect to the data can be improved after an optimal selection of signaling perturbations and how we learn optimal logic models with minimal number of experiments. The methods are applied on signaling pathways in human liver cells and phosphoproteomics experimental data. Using 25% of the experiments, we obtained logical models with fitness scores (mean square error) 15% close to the ones obtained using all experiments, illustrating the impact that our approach can have on the design of experiments for efficient model calibration.

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.

Genome-wide analysis of the WW domain-containing protein genes in silkworm and their expansion in eukaryotes

WW domains are protein modules that mediate protein-protein interactions through recognition of proline-rich peptide motifs and phosphorylated serine/threonine-proline sites. WW domains are found in many different structural and signaling proteins that are involved in a variety of cellular processes. WW domain-containing proteins (WWCPs) and complexes have been implicated in major human diseases including cancer as well as in major signaling cascades such as the Hippo tumor suppressor pathway, making them targets for new diagnostics and therapeutics. There are a number of reports about the WWCPs in different species, but systematic analysis of the WWCP genes and its ligands is still lacking in silkworm and the other organisms. In this study, WWCP genes and PY motif-containing proteins have been identified and analyzed in 56 species including silkworm. Whole-genome screening of B. mori identified thirty-three proteins with thirty-nine WW domains located on thirteen chromosomes. In the 39 silkworm WW domains, 15 domains belong to the Group I WW domain; 14 domains were in Group II/III, 9 domains derived from 8 silkworm WWCPs could not be classified into any group, and Group IV contains only one WW domain. Based on gene annotation, silkworm WWCP genes have functions in multi-biology processes. A detailed list of WWCPs from the other 55 species was sorted in this work. In 14,623 silkworm predicted proteins, nearly 18 % contained PY motif, nearly 30 % contained various motifs totally that could be recognized by WW domains. Gene Ontology and KEGG analysis revealed that dozens of WW domain-binding proteins are involved in Wnt, Hedgehog, Notch, mTOR, EGF and Jak-STAT signaling pathway. Tissue expression patterns of WWCP genes and potential WWCP-binding protein genes on the third day of the fifth instar (L5D3) were examined by microarray analysis. Tissue expression profile analysis found that several WWCP genes and poly-proline or PY motif-containing protein genes took tissue- or gender-dependent expression manner in silkworms. We further analyzed WWCPs and PY motif-containing proteins in representative organisms of invertebrates and vertebrates. The results showed that there are no less than 16 and up to 29 WWCPs in insects, the average is 22. The number of WW domains in insects is no less than 19, and up to 47, the average is 36. In vertebrates, excluding the Hydrobiontes, the number of WWCPs is no less than 34 and up to 49, the average is 43. The number of WW domains in vertebrates is no less than 56 and up to 85, the average is 73. Phylogenetic analysis revealed that most homologous genes of the WWCP subfamily in vertebrates were duplicated during evolution and functions diverged. Nearly 1,000 PY motif-containing protein genes were found in insect genomes and nearly 2,000 genes in vertebrates. The different distributions of WWCP genes and PY motif-containing protein genes in different species revealed a possible positive correlation with organism complexity. In conclusion, this comprehensive bio-information analysis of WWCPs and its binding ligands would provide rich fundamental knowledge and useful information for further exploration of the function of the WW domain-containing proteins not only in silkworm, but also in other species.

Genetic networks governing heart development

Animal genomes contain a code for construction of the body plan from a fertilized egg. Understanding how genome information is deciphered to create the complex multilayered regulatory systems that drive organismal development, and which become altered in disease, is one of the greatest challenges in the biological sciences. The development of methods that effectively represent and communicate the complexity inherent in gene regulatory networks remains a major barrier. This review introduces the philosophy of systems biology and discusses recent progress in understanding the development of the heart at a systems biology level.