Describing the kinetics of polymer reactions, such as the formation of loops and hairpins in nucleic acids or polypeptides, is complicated by the structural dynamics of their chains. Although both intramolecular reactions, such as cyclization, and intermolecular reactions have been studied extensively, both experimentally and theoretically, there is to date no exact explicit analytical treatment of transport-limited polymer reaction kinetics, even in the case of the simplest (Rouse) model of monomers connected by linear springs. We introduce a new analytical approach to calculate the mean reaction time of polymer reactions that encompasses the non-Markovian dynamics of monomer motion. This requires that the conformational statistics of the polymer at the very instant of reaction be determined, which provides, as a by-product, new information on the reaction path. We show that the typical reactive conformation of the polymer is more extended than the equilibrium conformation, which leads to reaction times significantly shorter than predicted by the existing classical Markovian theory.

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...

Spontaneous mechanical oscillations occur in various types of biological systems where groups of motor molecules are elastically coupled to their environment. By using an optical trap to oppose the gliding motion of a single bead-tailed actin filament over a substrate densely coated with myosin motors, we mimicked this condition in vitro. We show that this minimal actomyosin system can oscillate spontaneously. Our finding accords quantitatively with a general theoretical framework where oscillatory instabilities emerge generically from the collective dynamics of molecular motors under load.

Simple ingredients, such as well-defined interactions and couplings for the velocity and orientation of self-propelled objects, are sufficient to produce complex collective behavior in assemblies of such entities. Here, we use assemblies of rodlike robots made motile through self-vibration. When confined in circular arenas, dilute assemblies of these rods act as a gas. Increasing the surface fraction leads to a collective behavior near the boundaries: polar clusters emerge while, in the bulk, gaslike behavior is retained. The coexistence between a gas and surface clusters is a direct consequence of inertial effects as shown by our simulations. A theoretical model, based on surface mediated transport accounts for this coexistence and illustrates the exact role of the boundaries. Our study paves the way towards the control of collective behavior: By using deformable but free to move arenas, we demonstrate that the surface induced clusters can lead to directed motion, while the topology of the surface states can be controlled by biasing the motility of the particles.