This paper proposes a discrete-time Extremum-Seeking Control (ESC) strategy based on the Super-Twisting algorithm to optimize online a microbial electrolysis cell (MEC) for green hydrogen production. Besides, the digital architecture of the ESC strategy is designed and embedded in an FPGA to minimize hardware and energy consumption. Closed-loop simulations on the MEC model demonstrate the feasibility of the discrete-time ESC strategy. Results show that hydrogen production is correctly maximized for different inlet substrate concentrations, and the FPGA requires an estimate of only 122mW.

In this paper, a very simple discrete-time extremum-seeking control strategy is proposed from the classical direct search gradient-based optimization algorithm. First, the optimization problem to solve and solvability conditions are presented. Then, the discrete-time extremum-seeking strategy is developed and its convergence is assured by demonstrating that the optimization error dynamics is globally asymptotically stable. Finally, the feasibility of the proposed extremum-seeking algorithm is demonstrated by simulations on a fermentation reactor for bioethanol 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.

Calcium ions act as messengers in a broad range of processes such as learning, apoptosis, and muscular movement. The transient profile and the temporal accumulation of calcium signals have been suggested as the two main characteristics in which calcium cues encode messages to be forwarded to downstream pathways. We address the analytical quantification of calcium temporal-accumulation in a long, thin section of a nonexcitable cell by solving a boundary value problem. In these expressions we note that the cytosolic Ca(2+) accumulation is independent of every intracellular calcium flux and depends on the Ca(2+) exchange across the membrane, cytosolic calcium diffusion, geometry of the cell, extracellular calcium perturbation, and initial concentrations. In particular, we analyse the time-integrated response of cytosolic calcium due to i) a localised initial concentration of cytosolic calcium and ii) transient extracellular perturbation of calcium. In these scenarios, we conclude that i) the range of calcium progression is confined to the vicinity of the initial concentration, thereby creating calcium microdomains; and ii) we observe a low-pass filtering effect in the response driven by extracellular Ca(2+) perturbations. Additionally, we note that our methodology can be used to analyse a broader range of stimuli and scenarios.