The first-passage time (FPT), defined as the time a random walker takes to reach a target point in a confining domain, is a key quantity in the theory of stochastic processes 1 . Its importance comes from its crucial role to quantify the efficiency of processes as varied as diffusion-limited reactions 2,3 , target search processes 4 or spreading of diseases 5 . Most methods to determine the FPT properties in confined domains have been limited to Markovian (memoryless) processes 3,6,7 . However, as soon as the random walker interacts with its environment, memory effects can not be neglected. Examples of non Markovian dynamics include single-file diffusion in narrow channels8 or the motion of a tracer particle either attached to a polymeric chain 9 or diffusing in simple 10 or complex fluids such as nematics 11 , dense soft colloids 12 or viscoelastic solutions 13,14 . Here, we introduce an analytical approach to calculate, in the limit of a large confining volume, the mean FPT of a Gaussian non-Markovian random walker to a target point. The non-Markovian features of the dynamics are encompassed by determining the statistical properties of the trajectory of the random walker in the future of the first-passage event, which are shown to govern the FPT kinetics. This analysis is applicable to a broad range of stochastic processes, possibly correlated at long-times. Our theoretical predictions are confirmed by numerical simulations for several examples of non-Markovian processes including the emblematic case of the Fractional Brownian Motion in one or higher dimensions. These results show, on the basis of Gaussian processes, the importance of memory effects in first-passage statistics of non-Markovian random walkers in confinement.It has long been recognized that the kinetics of reactions is influenced by the properties of the transport process that brings reactants into contact 1,2 . Transport can even be the rate limiting step and in this diffusion controlled regime, More precisely, we consider a non-Markovian Gaussian stochastic process x(t), defined in unconfined space, which represents the position of a random walker at time t, starting from x 0 at t = 0. As the process is non-Markovian, the FPT statistics in fact depends also on x(t) for t < 0. For the sake of simplicity, we assume that at t = 0 the process of constant average x 0 is in stationary state (see SI for more general initial conditions), with incrementsThe process x(t) is then entirely characterized by its Mean Square DisplacementSuch a quantity is routinely measured in single particle tracking experiments and in fact includes all the memory effects in the case of Gaussian processes. At long times, the MSD is assumed to diverge and thus, typically, the particle does not remain close to its initial position. Last, the process is continuous and non smooth 25 ( ẋ(t) 2 = +∞), meaning that the trajectory is irregular and of fractal type, similarly to the standard Brownian motion. Note that the class of random walks that we consider here covers a broad spe...

How living systems break symmetry in an organized manner is a fundamental question in biology. In wild-type Caenorhabditis elegans zygotes, symmetry breaking during anterior-posterior axis specification is guided by centrosomes, resulting in anterior-directed cortical flows and a single posterior PAR-2 domain. We uncover that C. elegans zygotes depleted of the Aurora A kinase AIR-1 or lacking centrosomes entirely usually establish two posterior PAR-2 domains, one at each pole. We demonstrate that AIR-1 prevents symmetry breaking early in the cell cycle, whereas centrosomal AIR-1 instructs polarity initiation thereafter. Using triangular microfabricated chambers, we establish that bipolarity of air-1(RNAi) embryos occurs effectively in a cell-shape and curvature-dependent manner. Furthermore, we develop an integrated physical description of symmetry breaking, wherein local PAR-2-dependent weakening of the actin cortex, together with mutual inhibition of anterior and posterior PAR proteins, provides a mechanism for spontaneous symmetry breaking without centrosomes.

It has been known for a long time that the kinetics of diffusion-limited reactions can be quantified by the time needed for a diffusing molecule to reach a target: the first-passage time (FPT). So far the general determination of the mean first-passage time to a target in confinement has left aside aging media, such as glassy materials, cellular media, or cold atoms in optical lattices. Here we consider general non-Markovian scale-invariant diffusion processes, which model a broad class of transport processes of molecules in aging media, and demonstrate that all the moments of the FPT obey universal scalings with the confining volume with nontrivial exponents. Our analysis shows that a nonlinear scaling with the volume of the mean FPT, which quantities the mean reaction time, is the hallmark of aging and provides a general tool to quantify its impact on reaction kinetics in confinement.