Morphogen gradients provide positional information during development. To uncover the minimal requirements for morphogen gradient formation, we have engineered a synthetic morphogen in Drosophila wing primordia. We show that an inert protein, green fluorescent protein (GFP), can form a detectable diffusion-based gradient in the presence of surface-associated anti-GFP nanobodies, which modulate the gradient by trapping the ligand and limiting leakage from the tissue. We next fused anti-GFP nanobodies to the receptors of Dpp, a natural morphogen, to render them responsive to extracellular GFP. In the presence of these engineered receptors, GFP could replace Dpp to organize patterning and growth in vivo. Concomitant expression of glycosylphosphatidylinositol (GPI)–anchored nonsignaling receptors further improved patterning, to near–wild-type quality. Theoretical arguments suggest that GPI anchorage could be important for these receptors to expand the gradient length scale while at the same time reducing leakage.

The rate of entropy production by a stochastic process quantifies how far it is from thermodynamic equilibrium. Equivalently, entropy production captures the degree to which global detailed balance and time-reversal symmetry are broken. Despite abundant references to entropy production in the literature and its many applications in the study of non-equilibrium stochastic particle systems, a comprehensive list of typical examples illustrating the fundamentals of entropy production is lacking. Here, we present a brief, self-contained review of entropy production and calculate it from first principles in a catalogue of exactly solvable setups, encompassing both discrete- and continuous-state Markov processes, as well as single- and multiple-particle systems. The examples covered in this work provide a stepping stone for further studies on entropy production of more complex systems, such as many-particle active matter, as well as a benchmark for the development of alternative mathematical formalisms.

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A positive rate of entropy production at steady-state is a distinctive feature of truly non-equilibrium processes. Exact results, while being often limited to simple models, offer a unique opportunity to explore the thermodynamic features of these processes in full details. Here we derive analytical results for the steady-state rate of entropy production in single particle systems driven away from equilibrium by the fluctuations of an external potential of arbitrary shapes. Subsequently, we provide exact results for a diffusive particle in a harmonic trap whose potential stiffness varies in time according to both discrete and continuous Markov processes. In particular, studying the case of a fully intermittent potential allows us to introduce an effective model of stochastic resetting for which it is possible to obtain finite non-negative entropy production. Altogether, this work lays the foundation for a non-equilibrium thermodynamic theory of fluctuating potentials, with immediate applications to stochastic resetting processes, fluctuations in optical traps and fluctuating interactions in living systems.

Among observables characterizing the random exploration of a graph or lattice, the cover time, namely, the time to visit every site, continues to attract widespread interest. Much insight about cover times is gained by mapping to the (spaceless) coupon collector problem, which amounts to ignoring spatiotemporal correlations, and an early conjecture that the limiting cover time distribution of regular random walks on large lattices converges to the Gumbel distribution in d 3 was recently proved rigorously. Furthermore, a number of mathematical and numerical studies point to the robustness of the Gumbel universality to modifications of the spatial features of the random search processes (e.g., introducing persistence and/or intermittence, or changing the graph topology). Here we investigate the robustness of the Gumbel universality to dynamical modification of the temporal features of the search, specifically by allowing the random walker to "accelerate" or "decelerate" upon visiting a previously unexplored site. We generalize the mapping mentioned above by relating the statistics of cover times to the roughness of 1/ f α Gaussian signals, leading to the conjecture that the Gumbel distribution is but one of a family of cover time distributions, ranging from Gaussian for highly accelerated cover, to exponential for highly decelerated cover. While our conjecture is confirmed by systematic Monte Carlo simulations in dimensions d > 3, our results for acceleration in d = 3 challenge the current understanding of the role of correlations in the cover time problem.

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Locomotion characteristics are often recorded within bounded spaces, a constraint which introduces geometry-specific biases and potentially complicates the inference of behavioural features from empirical observations. We describe how statistical properties of an uncorrelated random walk, namely the steady-state stopping location probability density and the empirical step probability density, are affected by enclosure in a bounded space. The random walk here is considered as a null model for an organism moving intermittently in such a space, that is, the points represent stopping locations and the step is the displacement between them. Closed-form expressions are derived for motion in one dimension and simple two-dimensional geometries, in addition to an implicit expression for arbitrary (convex) geometries. For the particular choice of no-go boundary conditions, we demonstrate that the empirical step distribution is related to the intrinsic step distribution, i.e. the one we would observe in unbounded space, via a multiplicative transformation dependent solely on the boundary geometry. This conclusion allows in practice for the compensation of boundary effects and the reconstruction of the intrinsic step distribution from empirical observations.

We introduce a procedure to test a theory for point particle entity, that is, whether said theory takes into account the discrete nature of the constituents of the system. We then identify the mechanism whereby particle entity is enforced in the context of two field-theoretic frameworks designed to incorporate the particle nature of the degrees of freedom, namely the Doi-Peliti field theory and the response field field theory that derives from Dean’s equation. While the Doi-Peliti field theory encodes the particle nature at a very fundamental level that is easily revealed, demonstrating the same for Dean’s equation is more involved and results in a number of surprising diagrammatic identities. We derive those and discuss their implications. These results are particularly pertinent in the context of active matter, whose surprising and often counterintuitive phenomenology rests wholly on the particle nature of the agents and their degrees of freedom as particles.