Efficient manipulation and generation of Kirchhoff polynomials for the analysis of non-equilibrium biochemical reaction networks.

Kirchhoff polynomials are central for deriving symbolic steady-state expressions of models whose dynamics are governed by linear diffusion on graphs. In biology, such models have been unified under a common linear framework subsuming studies across areas such as enzyme kinetics, G-protein coupled receptors, ion channels and gene regulation. Due to 'history dependence' away from thermodynamic equilibrium, these models suffer from a (super) exponential growth in the size of their symbolic steady-state expressions and, respectively, Kirchhoff polynomials. This algebraic explosion has limited applications of the linear framework. However, recent results on the graph-based prime factorization of Kirchhoff polynomials may help subdue the combinatorial complexity. By prime decomposing the graphs contained in an expression of Kirchhoff polynomials and identifying the graphs giving rise to equal polynomials, we formulate a coarse-grained variant of the expression suitable for symbolic simplification. We devise criteria to efficiently test the equality of Kirchhoff polynomials and propose two heuristic algorithms to explicitly generate individual Kirchhoff polynomials in a compressed form; they are inspired by algebraic simplifications but operate on the corresponding graphs. We illustrate the practicality of the developed theory and algorithms for a diverse set of graphs of different sizes and for non-equilibrium gene regulation analyses.

BioSwitch: a tool for the detection of bistability and multi-steady state behaviour in signalling and gene regulatory networks.

MOTIVATION: Multi-steady state behaviour, and in particular multi-stability, provides biological systems with the capacity to take reliable decisions (such as cell fate determination). A problem arising frequently in systems biology is to elucidate whether a signal transduction mechanism or a gene regulatory network has the capacity for multi-steady state behaviour, and consequently for a switch-like response to stimuli. Bifurcation diagrams are a powerful instrument in non-linear analysis to study the qualitative and quantitative behaviour of equilibria including bifurcation into different equilibrium branches and bistability. However, in the context of signalling pathways, the inherent large parametric uncertainty hampers the (direct) use of standard bifurcation tools. RESULTS: We present BioSwitch, a toolbox to detect multi-steady state behaviour in signalling pathways and gene regulatory networks. The tool combines results from chemical reaction network theory with global optimization to efficiently detect whether a signalling pathway has the capacity to undergo a saddle node bifurcation, and in case of multi-stationarity, provides the exact coordinates of the bifurcation where to start a numerical continuation analysis with standard bifurcation tools, leading to two different branches of equilibria. Bistability detection in the G1/S transition pathway of Saccharomyces cerevisiae is included as an illustrative example. AVAILABILITY AND IMPLEMENTATION: BioSwitch runs under the popular MATLAB computational environment, and is available at https://sites.google.com/view/bioswitch.

Steady-State Differential Dose Response in Biological Systems.

In pharmacology and systems biology, it is a fundamental problem to determine how biological systems change their dose-response behavior upon perturbations. In particular, it is unclear how topologies, reactions, and parameters (differentially) affect the dose response. Because parameters are often unknown, systematic approaches should directly relate network structure and function. Here, we outline a procedure to compare general non-monotone dose-response curves and subsequently develop a comprehensive theory for differential dose responses of biochemical networks captured by non-equilibrium steady-state linear framework models. Although these models are amenable to analytical derivations of non-equilibrium steady states in principle, their size frequently increases (super) exponentially with model size. We extract general principles of differential responses based on a model's graph structure and thereby alleviate the combinatorial explosion. This allows us, for example, to determine reactions that affect differential responses, to identify classes of networks with equivalent differential, and to reject hypothetical models reliably without needing to know parameter values. We exemplify such applications for models of insulin signaling.

Chemical Reaction Network Theory elucidates sources of multistability in interferon signaling.

Bistability has important implications in signaling pathways, since it indicates a potential cell decision between alternative outcomes. We present two approaches developed in the framework of the Chemical Reaction Network Theory for easy and efficient search of multiple steady state behavior in signaling networks (both with and without mass conservation), and apply them to search for sources of bistability at different levels of the interferon signaling pathway. Different type I interferon subtypes and/or doses are known to elicit differential bioactivities (ranging from antiviral, antiproliferative to immunomodulatory activities). How different signaling outcomes can be generated through the same receptor and activating the same JAK/STAT pathway is still an open question. Here, we detect bistability at the level of early STAT signaling, showing how two different cell outcomes are achieved under or above a threshold in ligand dose or ligand-receptor affinity. This finding could contribute to explain the differential signaling (antiviral vs apoptotic) depending on interferon dose and subtype (α vs ß) observed in type I interferons.

A method for inverse bifurcation of biochemical switches: inferring parameters from dose response curves.

BACKGROUND: Within cells, stimuli are transduced into cell responses by complex networks of biochemical reactions. In many cell decision processes the underlying networks behave as bistable switches, converting graded stimuli or inputs into all or none cell responses. Observing how systems respond to different perturbations, insight can be gained into the underlying molecular mechanisms by developing mathematical models. Emergent properties of systems, like bistability, can be exploited to this purpose. One of the main challenges in modeling intracellular processes, from signaling pathways to gene regulatory networks, is to deal with high structural and parametric uncertainty, due to the complexity of the systems and the difficulty to obtain experimental measurements. Formal methods that exploit structural properties of networks for parameter estimation can help to overcome these problems. RESULTS: We here propose a novel method to infer the kinetic parameters of bistable biochemical network models. Bistable systems typically show hysteretic dose response curves, in which the so called bifurcation points can be located experimentally. We exploit the fact that, at the bifurcation points, a condition for multistationarity derived in the context of the Chemical Reaction Network Theory must be fulfilled. Chemical Reaction Network Theory has attracted attention from the (systems) biology community since it connects the structure of biochemical reaction networks to qualitative properties of the corresponding model of ordinary differential equations. The inverse bifurcation method developed here allows determining the parameters that produce the expected behavior of the dose response curves and, in particular, the observed location of the bifurcation points given by experimental data. CONCLUSIONS: Our inverse bifurcation method exploits inherent structural properties of bistable switches in order to estimate kinetic parameters of bistable biochemical networks, opening a promising route for developments in Chemical Reaction Network Theory towards kinetic model identification.