A rule-based multiscale model of hepatic stellate cell plasticity: Critical role of the inactivation loop in fibrosis progression.

Hepatic stellate cells (HSC) are the source of extracellular matrix (ECM) whose overproduction leads to fibrosis, a condition that impairs liver functions in chronic liver diseases. Understanding the dynamics of HSCs will provide insights needed to develop new therapeutic approaches. Few models of hepatic fibrosis have been proposed, and none of them include the heterogeneity of HSC phenotypes recently highlighted by single-cell RNA sequencing analyses. Here, we developed rule-based models to study HSC dynamics during fibrosis progression and reversion. We used the Kappa graph rewriting language, for which we used tokens and counters to overcome temporal explosion. HSCs are modeled as agents that present seven physiological cellular states and that interact with (TGFß1) molecules which regulate HSC activation and the secretion of type I collagen, the main component of the ECM. Simulation studies revealed the critical role of the HSC inactivation process during fibrosis progression and reversion. While inactivation allows elimination of activated HSCs during reversion steps, reactivation loops of inactivated HSCs (iHSCs) are required to sustain fibrosis. Furthermore, we demonstrated the model's sensitivity to (TGFß1) parameters, suggesting its adaptability to a variety of pathophysiological conditions for which levels of (TGFß1) production associated with the inflammatory response differ. Using new experimental data from a mouse model of CCl4-induced liver fibrosis, we validated the predicted ECM dynamics. Our model also predicts the accumulation of iHSCs during chronic liver disease. By analyzing RNA sequencing data from patients with non-alcoholic steatohepatitis (NASH) associated with liver fibrosis, we confirmed this accumulation, identifying iHSCs as novel markers of fibrosis progression. Overall, our study provides the first model of HSC dynamics in chronic liver disease that can be used to explore the regulatory role of iHSCs in liver homeostasis. Moreover, our model can also be generalized to fibroblasts during repair and fibrosis in other tissues.

The Kappa platform for rule-based modeling.

Motivation: We present an overview of the Kappa platform, an integrated suite of analysis and visualization techniques for building and interactively exploring rule-based models. The main components of the platform are the Kappa Simulator, the Kappa Static Analyzer and the Kappa Story Extractor. In addition to these components, we describe the Kappa User Interface, which includes a range of interactive visualization tools for rule-based models needed to make sense of the complexity of biological systems. We argue that, in this approach, modeling is akin to programming and can likewise benefit from an integrated development environment. Our platform is a step in this direction. Results: We discuss details about the computation and rendering of static, dynamic, and causal views of a model, which include the contact map (CM), snaphots at different resolutions, the dynamic influence network (DIN) and causal compression. We provide use cases illustrating how these concepts generate insight. Specifically, we show how the CM and snapshots provide information about systems capable of polymerization, such as Wnt signaling. A well-understood model of the KaiABC oscillator, translated into Kappa from the literature, is deployed to demonstrate the DIN and its use in understanding systems dynamics. Finally, we discuss how pathways might be discovered or recovered from a rule-based model by means of causal compression, as exemplified for early events in EGF signaling. Availability and implementation: The Kappa platform is available via the project website at kappalanguage.org. All components of the platform are open source and freely available through the authors' code repositories.

Internal coarse-graining of molecular systems.

Modelers of molecular signaling networks must cope with the combinatorial explosion of protein states generated by posttranslational modifications and complex formation. Rule-based models provide a powerful alternative to approaches that require explicit enumeration of all possible molecular species of a system. Such models consist of formal rules stipulating the (partial) contexts wherein specific protein-protein interactions occur. These contexts specify molecular patterns that are usually less detailed than molecular species. Yet, the execution of rule-based dynamics requires stochastic simulation, which can be very costly. It thus appears desirable to convert a rule-based model into a reduced system of differential equations by exploiting the granularity at which rules specify interactions. We present a formal (and automated) method for constructing a coarse-grained and self-consistent dynamical system aimed at molecular patterns that are distinguishable by the dynamics of the original system as posited by the rules. The method is formally sound and never requires the execution of the rule-based model. The coarse-grained variables do not depend on the values of the rate constants appearing in the rules, and typically form a system of greatly reduced dimension that can be amenable to numerical integration and further model reduction techniques.

Intrinsic information carriers in combinatorial dynamical systems.

Many proteins are composed of structural and chemical features--"sites" for short--characterized by definite interaction capabilities, such as noncovalent binding or covalent modification of other proteins. This modularity allows for varying degrees of independence, as the behavior of a site might be controlled by the state of some but not all sites of the ambient protein. Independence quickly generates a startling combinatorial complexity that shapes most biological networks, such as mammalian signaling systems, and effectively prevents their study in terms of kinetic equations-unless the complexity is radically trimmed. Yet, if combinatorial complexity is key to the system's behavior, eliminating it will prevent, not facilitate, understanding. A more adequate representation of a combinatorial system is provided by a graph-based framework of rewrite rules where each rule specifies only the information that an interaction mechanism depends on. Unlike reactions, which deal with molecular species, rules deal with patterns, i.e., multisets of molecular species. Although the stochastic dynamics induced by a collection of rules on a mixture of molecules can be simulated, it appears useful to capture the system's average or deterministic behavior by means of differential equations. However, expansion of the rules into kinetic equations at the level of molecular species is not only impractical, but conceptually indefensible. If rules describe bona fide patterns of interaction, molecular species are unlikely to constitute appropriate units of dynamics. Rather, we must seek aggregate variables reflective of the causal structure laid down by the rules. We call these variables "fragments" and the process of identifying them "fragmentation." Ideally, fragments are aspects of the system's microscopic population that the set of rules can actually distinguish on average; in practice, it may only be feasible to identify an approximation to this. Most importantly, fragments are self-consistent descriptors of system dynamics in that their time-evolution is governed by a closed system of kinetic equations. Taken together, fragments are endogenous distinctions that matter for the dynamics of a system, which warrants viewing them as the carriers of information. Although fragments can be thought of as multisets of molecular species (an extensional view), their self-consistency suggests treating them as autonomous aspects cut off from their microscopic realization (an intensional view). Fragmentation is a seeded process that depends on the choice of observables whose dynamics one insists to describe. Different observables can cause distinct fragmentations, in effect altering the set of information carriers that govern the behavior of a system, even though nothing has changed in its microscopic constitution. In this contribution, we present a mathematical specification of fragments, but not an algorithmic implementation. We have described the latter elsewhere in rather technical terms that, although effective, were lacking an embedding into a more general conceptual framework, which we here provide.

Local Traces: An Over-Approximation of the Behavior of the Proteins in Rule-Based Models.

Thanks to rule-based modelling languages, we can assemble large sets of mechanistic protein-protein interactions within integrated models. Our goal would be to understand how the behavior of these systems emerges from these low-level interactions. Yet, this is a quite long term challenge and it is desirable to offer intermediary levels of abstraction, so as to get a better understanding of the models and to increase our confidence within our mechanistic assumptions. To this extend, static analysis can be used to derive various abstractions of the semantics, each of them offering new perspectives on the models. We propose an abstract interpretation of the behavior of each protein, in isolation. Given a model written in Kappa, this abstraction computes for each kind of proteins a transition system that describes which conformations this protein may take and how a protein may pass from one conformation to another one. Then, we use simplicial complexes to abstract away the interleaving order of the transformations between conformations that commute. As a result, we get a compact summary of the potential behavior of each protein of the model.

Formal derivation of qualitative dynamical models from biochemical networks.

As technological advances allow a better identification of cellular networks, large-scale molecular data are swiftly produced, allowing the construction of large and detailed molecular interaction maps. One approach to unravel the dynamical properties of such complex systems consists in deriving coarse-grained dynamical models from these maps, which would make the salient properties emerge. We present here a method to automatically derive such models, relying on the abstract interpretation framework to formally relate model behaviour at different levels of description. We illustrate our approach on two relevant case studies: the formation of a complex involving a protein adaptor, and a race between two competing biochemical reactions. States and traces of reaction networks are first abstracted by sampling the number of instances of chemical species within a finite set of intervals. We show that the qualitative models induced by this abstraction are too coarse to reproduce properties of interest. We then refine our approach by taking into account additional constraints, the mass invariants and the limiting resources for interval crossing, and by introducing information on the reaction kinetics. The resulting qualitative models are able to capture sophisticated properties of interest, such as a sequestration effect, which arise in the case studies and, more generally, participate in shaping the dynamics of cell signaling and regulatory networks. Our methodology offers new trade-offs between complexity and accuracy, and clarifies the implicit assumptions made in the process of qualitative modelling of biological networks.

Combinatorial complexity and compositional drift in protein interaction networks.

The assembly of molecular machines and transient signaling complexes does not typically occur under circumstances in which the appropriate proteins are isolated from all others present in the cell. Rather, assembly must proceed in the context of large-scale protein-protein interaction (PPI) networks that are characterized both by conflict and combinatorial complexity. Conflict refers to the fact that protein interfaces can often bind many different partners in a mutually exclusive way, while combinatorial complexity refers to the explosion in the number of distinct complexes that can be formed by a network of binding possibilities. Using computational models, we explore the consequences of these characteristics for the global dynamics of a PPI network based on highly curated yeast two-hybrid data. The limited molecular context represented in this data-type translates formally into an assumption of independent binding sites for each protein. The challenge of avoiding the explicit enumeration of the astronomically many possibilities for complex formation is met by a rule-based approach to kinetic modeling. Despite imposing global biophysical constraints, we find that initially identical simulations rapidly diverge in the space of molecular possibilities, eventually sampling disjoint sets of large complexes. We refer to this phenomenon as "compositional drift". Since interaction data in PPI networks lack detailed information about geometric and biological constraints, our study does not represent a quantitative description of cellular dynamics. Rather, our work brings to light a fundamental problem (the control of compositional drift) that must be solved by mechanisms of assembly in the context of large networks. In cases where drift is not (or cannot be) completely controlled by the cell, this phenomenon could constitute a novel source of phenotypic heterogeneity in cell populations.