How large the number of redundant copies should be to make a rare event probable.

The redundancy principle provides a framework to study how rare events are made possible with probability 1 in accelerated time, by making many copies of similar random searchers. However, what is a large n? To estimate large n with respect to the geometrical properties of a domain and the dynamics, we present here a criterion based on splitting probabilities between a small fraction of the exploration space associated with an activation process and other absorbing regions where trajectories can be terminated. We obtain explicit computations especially when there is a killing region located inside the domain that we compare with stochastic simulations. We also present examples of extreme trajectories with killing in dimension 2. For a large n, the optimal trajectories avoid penetrating inside the killing region. Finally, we discuss some applications to cell biology.

Pattern forming systems coupling linear bulk diffusion to dynamically active membranes or cells.

Some analytical and numerical results are presented for pattern formation properties associated with novel types of reaction-diffusion (RD) systems that involve the coupling of bulk diffusion in the interior of a multi-dimensional spatial domain to nonlinear processes that occur either on the domain boundary or within localized compartments that are confined within the domain. The class of bulk-membrane system considered herein is derived from an asymptotic analysis in the limit of small thickness of a thin domain that surrounds the bulk medium. When the bulk domain is a two-dimensional disk, a weakly nonlinear analysis is used to characterize Turing and Hopf bifurcations that can arise from the linearization around a radially symmetric, but spatially non-uniform, steady-state of the bulk-membrane system. In a singularly perturbed limit, the existence and linear stability of localized membrane-bound spike patterns is analysed for a Gierer-Meinhardt activator-inhibitor model that includes bulk coupling. Finally, the emergence of collective intracellular oscillations is studied for a class of PDE-ODE bulk-cell model in a bounded two-dimensional domain that contains spatially localized, but dynamically active, circular cells that are coupled through a linear bulk diffusion field. Applications of such coupled bulk-membrane or bulk-cell systems to some biological systems are outlined, and some open problems in this area are discussed. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.