Abstract
AbstractWe study the minimal‐time problem for a piecewise affine bistable switch. Motivated by applications in synthetic biology and biotechnology, the aim is to minimize the time needed for this system to achieve transitions between its two stable steady states. The latter represents the two possible states of a genetic toggle switch, a synthetic flip‐flop device playing a fundamental role in biocomputing and gene therapy. Results show that a time‐optimal transition between states should pass by an undifferentiated state, which is well known in cell biology for its importance in fate differentiation of cells. In order to characterize the capacity of the system to achieve transitions, we provide a lower bound on the minimal time, whose knowledge becomes relevant when considering realistic systems involving subsystems evolving on different time scales. Then, we show numerical simulations of optimal trajectories illustrating the structure of the bang‐bang optimal control for different scenarios.
Highlights
Understanding complex biological phenomena has become of great interest in the last decades for the scientific community
It often occurs that trajectories belonging to a given domain may bifurcate into different domains, to what happens in the toggle switch case, and some similar turnpike-like properties may hold in this case
Through the application of HMP, we showed that any optimal control achieving state transition is a bang-bang control, where its value is a function of the state of the system
Summary
Understanding complex biological phenomena has become of great interest in the last decades for the scientific community. The simplest positive feedback loop is the two-dimensional bistable system, which is commonly used to represent the so-called genetic toggle switch The latter is a synthetic flip-flop device first implemented experimentally in E. coli through the genes lacI and tetR mutually repressing each other [13]. One of the key issues in these genetic devices is the time needed to induce a transfer between its two stable states, due to its importance when studying more complex networks of systems involving different time scales The latter becomes a major constraint in the framework of biological signal processing [24]. We perform a numerical comparison between the trajectories of the relaxed OCP and the original OCP, that suggests that the results hold for the original one
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