Equilibria and stability of a class of positive feedback loops: Mathematical analysis and its application to caspase-dependent apoptosis

Stable IL- 1β-Activation in an Inflammasome Signalling Model Depends on Positive and Negative Feedbacks and Tight Regulation of Protein Production

INTRODUCTION

NLRP3-dependent inflammasome signalling is a key pathway during inflammatory processes and its deregulation is implicated in several diseases. NLRP3-inflammasome pathway activation leads to the rapid, phosphorylation-driven NF$\kappa$κB-pathway signalling, subsequently proceeds via slower transcription/translation process for producing pro-enzymes, and finally leads to the medium-speed enzymatic activation of the central inflammatory mediator IL-$1\beta$1β[1] . We here were interested in how the timing of the rate-limiting step of transcription/translation and the presence of a positive and negative auto-regulation would pose conditions for meaningful and stable IL-$1\beta$1β-activation.

METHODS

We extracted the essential topology of the inflammasome pathway network using a linear chain of first-order reaction and a second-order reaction for inhibitory feedback. We then performed an analytical treatment of the resulting ODE set to obtain closed-form formulae. We therefore looked for the steady states and characterized their stability by using a Jacobian-based, local analysis. We employed the Small Gain Theorem from Control Theory as recently applied by us [2] and the Gershgorin Circle Theorem to obtain mathematically exact conditions for a positive ON state and stabilities for ON and OFF steady states.

RESULTS

We identified an ON- and one OFF- steady state whose properties we characterized in terms of the kinetic parameters by closed-form formulae. We found that under the assumption of a first-order information flow through the network, the existence of a biologically reasonable ON steady state required the simultaneous presence of the positive and the negative feedback. Assuming non-competitivity between IL-$1\beta$1β entities binding to different receptors, we found that a minimum kinetics for protein production is required to sustain a steady state with IL-$1\beta$1β activation. Assuming competitivity between IL-$1\beta$1β entities introduced additional restrictions on the maximum protein production speed to guarantee a biologically reasonable ON steady state. Finally, for both models, we ruled out bistability, suggesting that IL-$1\beta$1β activation would undergo a smooth change upon alterations of its parameters.

CONCLUSION

Exemplified by the core pathway of NLRP3-inflammasome signalling, we here demonstrate that a mostly linear activation cascade containing an intermediate rate limiting step poses kinetic restrictions on this step and requires positive and negative autoregulation for obtaining a meaningful ON steady state. Due to the generality of our framework, our results are important for a wide class of receptor mediated-pathways, where a fast initial phosphorylation cascade is followed by a (slower) transcriptional response and subsequent autoregulation. Our results may further provide important design principles for synthetic biological networks involving biochemical activation and transcription/translation, by relating timing considerations and autoregulation to stable pathway activation.

Anti-apoptotic potential of several antidiabetic medicinal plants of the eastern James Bay Cree pharmacopeia in cultured kidney cells

Background

Our team has identified 17 Boreal forest species from the traditional pharmacopeia of the Eastern James Bay Cree that presented promising in vitro and in vivo biological activities in the context of type 2 diabetes (T2D).

We now screened the 17 plants extracts for potential anti-apoptotic activity in cultured kidney cells and investigated the underlying mechanisms.

Methods

MDCK (Madin-Darnby Canine Kidney) cell damage was induced by hypertonic medium (700 mOsm/L) in the presence or absence of maximal nontoxic concentrations of each of the 17 plant extracts. After 18 h’ treatment, cells were stained with Annexin V (AnnV) and Propidium iodide (PI) and subjected to flow cytometry to assess the cytoprotective (AnnV⁻/PI⁻) and anti-apoptotic (AnnV⁺/PI⁻) potential of the 17 plant extracts. We then selected a representative subset of species (most cytoprotective, moderately so or neutral) to measure the activity of caspases 3, 8 and 9.

Results

Gaultheria hispidula and Abies balsamea are amongst the most powerful cytoprotective and anti-apoptotic plants and appear to exert their modulatory effect primarily by inhibiting caspase 9 in the mitochondrial apoptotic signaling pathway.

Conclusion

We conclude that several Cree antidiabetic plants exert anti-apoptotic activity that may be relevant in the context of diabetic nephropathy (DN) that affects a significant proportion of Cree diabetics.

Exact probability distributions of selected species in stochastic chemical reaction networks

Chemical reactions are discrete, stochastic events. As such, the species' molecular numbers can be described by an associated master equation. However, handling such an equation may become difficult due to the large size of reaction networks. A commonly used approach to forecast the behaviour of reaction networks is to perform computational simulations of such systems and analyse their outcome statistically. This approach, however, might require high computational costs to provide accurate results. In this paper we opt for an analytical approach to obtain the time-dependent solution of the Chemical Master Equation for selected species in a general reaction network. When the reaction networks are composed exclusively of zeroth and first-order reactions, this analytical approach significantly alleviates the computational burden required by simulation-based methods. By building upon these analytical solutions, we analyse a general monomolecular reaction network with an arbitrary number of species to obtain the exact marginal probability distribution for selected species. Additionally, we study two particular topologies of monomolecular reaction networks, namely (i) an unbranched chain of monomolecular reactions with and without synthesis and degradation reactions and (ii) a circular chain of monomolecular reactions. We illustrate our methodology and alternative ways to use it for non-linear systems by analysing a protein autoactivation mechanism. Later, we compare the computational load required for the implementation of our results and a pure computational approach to analyse an unbranched chain of monomolecular reactions. Finally, we study calcium ions gates in the sarco/endoplasmic reticulum mediated by ryanodine receptors.

Cyclic negative feedback systems: what is the chance of oscillation?

Many biological oscillators have a cyclic structure consisting of negative feedback loops. In this paper, we analyze the impact that the addition of a positive or a negative self-feedback loop has on the oscillatory behavior of the three negative feedback oscillators proposed by Tsai et al. (Science 231:126-129, 2008) where, in contrast with numerous oscillator models, the interactions between elements occur via the modulation of the degradation rates. Through analytical and computational studies we show that an additional self-feedback affects the oscillatory behavior. In the high-cooperativity limit, i.e., for large Hill coefficients, we derive exact analytical conditions for oscillations and show that the relative location between the dissociation constants of the Hill functions and the ratio of kinetic parameters determines the possibility of oscillatory activities. We compute analytically the probability of oscillations for the three models and show that the smallest domain of periodic behavior is obtained for the negative-plus-negative feedback system whereas the additional positive self-feedback loop does not modify significantly the chance to oscillate. We numerically investigate to what extent the properties obtained in the sharp situation applied in the smooth case. Results suggest that a switch-like coupling behavior, a time-scale separation, and a repressilator-type architecture with an even number of elements facilitate the emergence of sustained oscillations in biological systems. An additional positive self-feedback loop produces robustness and adaptability whereas an additional negative self-feedback loop reduces the chance to oscillate.