A Reduction of Logical Regulatory Graphs Preserving Essential Dynamical Properties

Patient-Specific Modeling of Diffuse Large B-Cell Lymphoma

Personalized medicine aims to tailor treatment to patients based on their individual genetic or molecular background. Especially in diseases with a large molecular heterogeneity, such as diffuse large B-cell lymphoma (DLBCL), personalized medicine has the potential to improve outcome and/or to reduce resistance towards treatment. However, integration of patient-specific information into a computational model is challenging and has not been achieved for DLBCL. Here, we developed a computational model describing signaling pathways and expression of critical germinal center markers. The model integrates the regulatory mechanism of the signaling and gene expression network and covers more than 50 components, many carrying genetic lesions common in DLBCL. Using clinical and genomic data of 164 primary DLBCL patients, we implemented mutations, structural variants and copy number alterations as perturbations in the model using the CoLoMoTo notebook. Leveraging patient-specific genotypes and simulation of the expression of marker genes in specific germinal center conditions allows us to predict the consequence of the modeled pathways for each patient. Finally, besides modeling how genetic perturbations alter physiological signaling, we also predicted for each patient model the effect of rational inhibitors, such as Ibrutinib, that are currently discussed as possible DLBCL treatments, showing patient-dependent variations in effectiveness and synergies.

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.

Deciphering the expression dynamics of ANGPTL8 associated regulatory network in insulin resistance using formal modelling approaches

ANGPTL8 is a recently identified novel hormone which regulates both glucose and lipid metabolism. The increase in ANGPTL8 during compensatory insulin resistance has been recently reported to improve glucose tolerance and a part of cytoprotective metabolic circuit. However, the exact signalling entities and dynamics involved in this process have remained elusive. Therefore, the current study was conducted with a specific aim to model the regulation of ANGPTL8 with emphasis on its role in improving glucose tolerance during insulin resistance. The main contribution of this study is the construction of a discrete model (based on kinetic logic of René Thomas) and its equivalent Stochastic Petri Net model of ANGPTL8 associated Biological Regulatory Network (BRN) which can predict its dynamic behaviours. The predicted results of these models are in‐line with the previous experimental observations and provide comprehensive insights into the signalling dynamics of ANGPTL8 associated BRN. The authors’ results support the hypothesis that ANGPTL8 plays an important role in supplementing the insulin signalling pathway during insulin resistance and its loss can aggravate the pathogenic process by quickly leading towards Diabetes Mellitus. The results of this study have potential therapeutic implications for treatment of Diabetes Mellitus and are suggestive of its potential as a glucose‐lowering agent.

Identification of Biologically Essential Nodes via Determinative Power in Logical Models of Cellular Processes

A variety of biological networks can be modeled as logical or Boolean networks. However, a simplification of the reality to binary states of the nodes does not ease the difficulty of analyzing the dynamics of large, complex networks, such as signal transduction networks, due to the exponential dependence of the state space on the number of nodes. This paper considers a recently introduced method for finding a fairly small subnetwork, representing a collection of nodes that determine the states of most other nodes with a reasonable level of entropy. The subnetwork contains the most determinative nodes that yield the highest information gain. One of the goals of this paper is to propose an algorithm for finding a suitable subnetwork size. The information gain is quantified by the so-called determinative power of the nodes, which is obtained via the mutual information, a concept originating in information theory. We find the most determinative nodes for 36 network models available in the online database Cell Collective (http://cellcollective.org). We provide statistical information that indicates a weak correlation between the subnetwork size and other variables, such as network size, or maximum and average determinative power of nodes. We observe that the proportion represented by the subnetwork in comparison to the whole network shows a weak tendency to decrease for larger networks. The determinative power of nodes is weakly correlated to the number of outputs of a node, and it appears to be independent of other topological measures such as closeness or betweenness centrality. Once the subnetwork of the most determinative nodes is identified, we generate a biological function analysis of its nodes for some of the 36 networks. The analysis shows that a large fraction of the most determinative nodes are essential and involved in crucial biological functions. The biological pathway analysis of the most determinative nodes shows that they are involved in important disease pathways.

Influence Networks Compared with Reaction Networks: Semantics, Expressivity and Attractors

Biochemical reaction networks are one of the most widely used formalisms in systems biology to describe the molecular mechanisms of high-level cell processes. However, modellers also reason with influence diagrams to represent the positive and negative influences between molecular species and may find an influence network useful in the process of building a reaction network. In this paper, we introduce a formalism of influence networks with forces, and equip it with a hierarchy of Boolean, Petri net, stochastic and differential semantics, similarly to reaction networks with rates. We show that the expressive power of influence networks is the same as that of reaction networks under the differential semantics, but weaker under the discrete semantics. Furthermore, the hierarchy of semantics leads us to consider a (positive) Boolean semantics that cannot test the absence of a species, that we compare with the (negative) Boolean semantics with test for absence of a species in gene regulatory networks à la Thomas. We study the monotonicity properties of the positive semantics and derive from them an algorithm to compute attractors in both the positive and negative Boolean semantics. We illustrate our results on models of the literature about the p53/Mdm2 DNA damage repair system, the circadian clock, and the influence of MAPK signaling on cell-fate decision in urinary bladder cancer.

Maximal aggregation of polynomial dynamical systems

Ordinary differential equations (ODEs) with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science, but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models. Here we propose an aggregation technique that rests on two notions of equivalence relating ODE variables whenever they have the same solution (backward criterion) or if a self-consistent system can be written for describing the evolution of sums of variables in the same equivalence class (forward criterion). A key feature of our proposal is to encode a polynomial ODE system into a finitary structure akin to a formal chemical reaction network. This enables the development of a discrete algorithm to efficiently compute the largest equivalence, building on approaches rooted in computer science to minimize basic models of computation through iterative partition refinements. The physical interpretability of the aggregation is shown on polynomial ODE systems for biochemical reaction networks, gene regulatory networks, and evolutionary game theory.

Formal Modeling of mTOR Associated Biological Regulatory Network Reveals Novel Therapeutic Strategy for the Treatment of Cancer

Cellular homeostasis is a continuous phenomenon that if compromised can lead to several disorders including cancer. There is a need to understand the dynamics of cellular proliferation to get deeper insights into the prevalence of cancer. Mechanistic Target of Rapamycin (mTOR) is implicated as the central regulator of the metabolic pathway involved in growth whereas its two distinct complexes mTORC1 and mTORC2 perform particular functions in cellular propagation. To date, mTORC1 is a well defined therapeutic target to inhibit uncontrolled cell division, while the role of mTORC2 is not well characterized. Therefore, the current study is designed to understand the signaling dynamics of mTOR and its partner proteins such as PI3K, PTEN, mTORC2, PKB (Akt), mTORC1, and FOXO. For this purpose, a qualitative model of mTOR-associated Biological Regulatory Network (BRN) is constructed to predict its regulatory behaviors which may not be predictable otherwise. The depleted expression of PTEN and FOXO along with the overexpression of PI3K, mTORC2, mTORC1 and Akt is predicted as a stable steady state which is in accordance with their observed expression levels in the progression of various cancers. The qualitative model also predicts the homeostasis of all the entities in the form of qualitative cycles. The significant qualitative (discrete) cycle is identified by analyzing betweenness centralities of the qualitative (discrete) states. This cycle is further refined as a linear hybrid automaton model with the production (activation) and degradation (inhibition) time delays in order to analyze the real-time constraints for its existence. The analysis of the hybrid model provides a formal proof that during homeostasis the inhibition time delay of Akt is less than the inhibition time delay of mTORC2. In conclusion, our observations characterize that in homeostasis Akt is degraded with a faster rate than mTORC2 which suggests that the inhibition of Akt along with the activation of mTORC2 may be a better therapeutic strategy for the treatment of cancer.

Logical Reduction of Biological Networks to Their Most Determinative Components

Boolean networks have been widely used as models for gene regulatory networks, signal transduction networks, or neural networks, among many others. One of the main difficulties in analyzing the dynamics of a Boolean network and its sensitivity to perturbations or mutations is the fact that it grows exponentially with the number of nodes. Therefore, various approaches for simplifying the computations and reducing the network to a subset of relevant nodes have been proposed in the past few years. We consider a recently introduced method for reducing a Boolean network to its most determinative nodes that yield the highest information gain. The determinative power of a node is obtained by a summation of all mutual information quantities over all nodes having the chosen node as a common input, thus representing a measure of information gain obtained by the knowledge of the node under consideration. The determinative power of nodes has been considered in the literature under the assumption that the inputs are independent in which case one can use the Bahadur orthonormal basis. In this article, we relax that assumption and use a standard orthonormal basis instead. We use techniques of Hilbert space operators and harmonic analysis to generate formulas for the sensitivity to perturbations of nodes, quantified by the notions of influence, average sensitivity, and strength. Since we work on finite-dimensional spaces, our formulas and estimates can be and are formulated in plain matrix algebra terminology. We analyze the determinative power of nodes for a Boolean model of a signal transduction network of a generic fibroblast cell. We also show the similarities and differences induced by the alternative complete orthonormal basis used. Among the similarities, we mention the fact that the knowledge of the states of the most determinative nodes reduces the entropy or uncertainty of the overall network significantly. In a special case, we obtain a stronger result than in previous works, showing that a large information gain from a set of input nodes generates increased sensitivity to perturbations of those inputs.

Dynamical modeling of the cell cycle and cell fate emergence in Caulobacter crescentus

The division of Caulobacter crescentus, a model organism for studying cell cycle and differentiation in bacteria, generates two cell types: swarmer and stalked. To complete its cycle, C. crescentus must first differentiate from the swarmer to the stalked phenotype. An important regulator involved in this process is CtrA, which operates in a gene regulatory network and coordinates many of the interactions associated to the generation of cellular asymmetry. Gaining insight into how such a differentiation phenomenon arises and how network components interact to bring about cellular behavior and function demands mathematical models and simulations. In this work, we present a dynamical model based on a generalization of the Boolean abstraction of gene expression for a minimal network controlling the cell cycle and asymmetric cell division in C. crescentus. This network was constructed from data obtained from an exhaustive search in the literature. The results of the simulations based on our model show a cyclic attractor whose configurations can be made to correspond with the current knowledge of the activity of the regulators participating in the gene network during the cell cycle. Additionally, we found two point attractors that can be interpreted in terms of the network configurations directing the two cell types. The entire network is shown to be operating close to the critical regime, which means that it is robust enough to perturbations on dynamics of the network, but adaptable to environmental changes.

Formal modelling of toll like receptor 4 and JAK/STAT signalling pathways: insight into the roles of SOCS-1, interferon-β and proinflammatory cytokines in sepsis

Sepsis is one of the major causes of human morbidity and results in a considerable number of deaths each year. Lipopolysaccharide-induced sepsis has been associated with TLR4 signalling pathway which in collaboration with the JAK/STAT signalling regulate endotoxemia and inflammation. However, during sepsis our immune system cannot maintain a balance of cytokine levels and results in multiple organ damage and eventual death. Different opinions have been made in previous studies about the expression patterns and the role of proinflammatory cytokines in sepsis that attracted our attention towards qualitative properties of TLR4 and JAK/STAT signalling pathways using computer-aided studies. René Thomas' formalism was used to model septic and non-septic dynamics of TLR4 and JAK/STAT signalling. Comparisons among dynamics were made by intervening or removing the specific interactions among entities. Among our predictions, recurrent induction of proinflammatory cytokines with subsequent downregulation was found as the basic characteristic of septic model. This characteristic was found in agreement with previous experimental studies, which implicate that inflammation is followed by immunomodulation in septic patients. Moreover, intervention in downregulation of proinflammatory cytokines by SOCS-1 was found desirable to boost the immune responses. On the other hand, interventions either in TLR4 or transcriptional elements such as NFκB and STAT were found effective in the downregulation of immune responses. Whereas, IFN-β and SOCS-1 mediated downregulation at different levels of signalling were found to be associated with variations in the levels of proinflammatory cytokines. However, these predictions need to be further validated using wet laboratory experimental studies to further explore the roles of inhibitors such as SOCS-1 and IFN-β, which may alter the levels of proinflammatory cytokines at different stages of sepsis.

Logical modelling of gene regulatory networks with GINsim

Discrete mathematical formalisms are well adapted to model large biological networks, for which detailed kinetic data are scarce. This chapter introduces the reader to a well-established qualitative (logical) framework for the modelling of regulatory networks. Relying on GINsim, a software implementing this logical formalism, we guide the reader step by step towards the definition and the analysis of a simple model of the lysis-lysogeny decision in the bacteriophage λ.

Cell death and life in cancer: mathematical modeling of cell fate decisions

Tumor development is characterized by a compromised balance between cell life and death decision mechanisms, which are tightly regulated in normal cells. Understanding this process provides insights for developing new treatments for fighting with cancer. We present a study of a mathematical model describing cellular choice between survival and two alternative cell death modalities: apoptosis and necrosis. The model is implemented in discrete modeling formalism and allows to predict probabilities of having a particular cellular phenotype in response to engagement of cell death receptors. Using an original parameter sensitivity analysis developed for discrete dynamic systems, we determine variables that appear to be critical in the cellular fate decision and discuss how they are exploited by existing cancer therapies.

Mathematical modelling of cell-fate decision in response to death receptor engagement

Cytokines such as TNF and FASL can trigger death or survival depending on cell lines and cellular conditions. The mechanistic details of how a cell chooses among these cell fates are still unclear. The understanding of these processes is important since they are altered in many diseases, including cancer and AIDS. Using a discrete modelling formalism, we present a mathematical model of cell fate decision recapitulating and integrating the most consistent facts extracted from the literature. This model provides a generic high-level view of the interplays between NFkappaB pro-survival pathway, RIP1-dependent necrosis, and the apoptosis pathway in response to death receptor-mediated signals. Wild type simulations demonstrate robust segregation of cellular responses to receptor engagement. Model simulations recapitulate documented phenotypes of protein knockdowns and enable the prediction of the effects of novel knockdowns. In silico experiments simulate the outcomes following ligand removal at different stages, and suggest experimental approaches to further validate and specialise the model for particular cell types. We also propose a reduced conceptual model implementing the logic of the decision process. This analysis gives specific predictions regarding cross-talks between the three pathways, as well as the transient role of RIP1 protein in necrosis, and confirms the phenotypes of novel perturbations. Our wild type and mutant simulations provide novel insights to restore apoptosis in defective cells. The model analysis expands our understanding of how cell fate decision is made. Moreover, our current model can be used to assess contradictory or controversial data from the literature. Ultimately, it constitutes a valuable reasoning tool to delineate novel experiments.