Under nitrogen deprivation, the one-dimensional cyanobacterial organism Anabaena sp. PCC 7120 develops patterns of single, nitrogen-fixing cells separated by nearly regular intervals of photosynthetic vegetative cells. We study a minimal, stochastic model of developmental patterns in Anabaena that includes a nondiffusing activator, two diffusing inhibitor morphogens, demographic fluctuations in the number of morphogen molecules, and filament growth. By tracking developing filaments, we provide experimental evidence for different spatiotemporal roles of the two inhibitors during pattern maintenance and for small molecular copy numbers, justifying a stochastic approach. In the deterministic limit, the model yields Turing patterns within a region of parameter space that shrinks markedly as the inhibitor diffusivities become equal. Transient, noise-driven, stochastic Turing patterns are produced outside this region, which can then be fixed by downstream genetic commitment pathways, dramatically enhancing the robustness of pattern formation, also in the biologically relevant situation in which the inhibitors' diffusivities may be comparable.

We study the non-Markovian Langevin dynamics of a massive particle in a onedimensional double-well potential in the presence of multi-exponential memory by simulations. We consider memory functions as the sum of two or three exponentials with different friction amplitudes γi and different memory times τi and confirm the validity of a previously suggested heuristic formula for the mean first-passage time τMF P . Based on the heuristic formula, we derive a general scaling diagram that features a Markovian regime for short memory times, an asymptotic long-memory-time regime where barrier crossing is slowed down and τMF P grows quadratically with the memory time, and a non-Markovian intermediate regime where barrier crossing is slightly accelerated or slightly slowed down, depending primarily on the particle mass. The relative weight of different exponential memory contributions is described by the scaling variable γi/τ 2 i , i.e., memory contributions with long memory times or small amplitudes are negligible compared to other memory contributions.

Abstract. We investigate the mean-square displacement (MSD) for random motion governed by the generalized Langevin equation for memory functions that contain two different time scales: In the first model, the memory kernel consists of a delta peak and a single-exponential and in the second model of the sum of two exponentials. In particular, we investigate the scenario where the long-time exponential kernel contribution is negative. The competition between positive and negative friction memory contributions produces an enhanced transient persistent regime in the MSD, which is relevant for biological motility and active matter systems. Graphical abstract

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The non-equilibrium non-Markovian barrier-crossing dynamics of a one-dimensional massive coordinate, described by the non-equilibrium version of the generalized Langevin equation with unequal random and friction relaxation times, is studied by simulations and analytical methods. Within a harmonic approximation, a general formula for the barrier-crossing time is derived which agrees favorably with simulations. Non-equilibrium random forces with a relaxation time longer than the friction relaxation time induce non-Arrhenius behavior and dramatically increase the barrier-crossing time; within the harmonic theory this corresponds to a reduced effective temperature which also modifies the spatial and velocity distributions.