Summary The purpose of this paper is to present an extension of the generalised supertwisting algorithm (STA) to the multivariable framework. We begin by introducing an algorithm that may be deemed as a linear, quasicontinuous, or discontinuous multivariable system, depending on the functions that define them. For the class represented by such an algorithm we prove the robust, Lyapunov stability of the origin and characterise the perturbations that preserve its stability. In particular, when its vector field is discontinuous or quasicontinuous our algorithm is endowed with finite‐time stability. Due to its resemblance to the scalar case, we denote such finite‐time stable systems as generalised multivariable STA. Furthermore, the class of finite‐time stable systems comprise the currently available versions of STAs. To finalise, by means of simulation examples, we show that our proposed finite‐time stable algorithms are well suited for signals online differentiation and highlight their dynamical traits.

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Positive feedback loops are common regulatory elements in metabolic and protein signalling pathways. The length of such feedback loops determines stability and sensitivity to network perturbations. Here we provide a mathematical analysis of arbitrary length positive feedback loops with protein production and degradation. These loops serve as an abstraction of typical regulation patterns in protein signalling pathways. We first perform a steady state analysis and, independently of the chain length, identify exactly two steady states that represent either biological activity or inactivity. We thereby provide two formulas for the steady state protein concentrations as a function of feedback length, strength of feedback, as well as protein production and degradation rates. Using a control theory approach, analysing the frequency response of the linearisation of the system and exploiting the Small Gain Theorem, we provide conditions for local stability for both steady states. Our results demonstrate that, under some parameter relationships, once a biological meaningful on steady state arises, it is stable, while the off steady state, where all proteins are inactive, becomes unstable. We apply our results to a three-tier feedback of caspase activation in apoptosis and demonstrate how an intermediary protein in such a loop may be used as a signal amplifier within the cascade. Our results provide a rigorous mathematical analysis of positive feedback chains of arbitrary length, thereby relating pathway structure and stability.

Cells sense information encoded in extracellular ligand concentrations and process it using intracellular signalling cascades. Using mathematical modelling and high-throughput imaging of individual cells, we studied how a transient extracellular growth factor signal is sensed by the epidermal growth factor receptor system, processed by downstream signalling, and transmitted to the nucleus. We found that transient epidermal growth factor signals are linearly translated into an activated epidermal growth factor receptor integrated over time. This allows us to generate a simplified model of receptor signaling where the receptor acts as a perfect sensor of extracellular information, while the nonlinear input-output relationship of EGF-EGFR triggered signalling is a consequence of the downstream MAPK cascade alone. Insight, innovation, integrationWe derived an analytical formula from nonlinear models of receptor signalling that have been used to model the signalling of the epidermal growth factor receptor (EGFR). The formula predicted that the cumulative EGFR signalling is linearly dependent on the ligand concentration outside the cell. Quantitative imaging of HeLa cells suggests that the cumulative EGFR signalling indeed depends linearly on the EGF concentration. Our mathematical approach therefore allows a simplified view on complex networks of receptor dynamics and describes the EGFR as a linear sensor of extracellular information encoded in ligand concentrations.