Stable IL-$1\beta$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.