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We study the electro-diffusion properties of a domain containing a cusp-shaped structure in three dimensions when one ionic specie is dominant. The mathematical problem consists in solving the steady-state Poisson-Nernst-Planck (PNP) equation with an integral constraint for the number of charges. A non-homogeneous Neumann boundary condition is imposed on the boundary. We construct an asymptotic approximation for certain singular limits that agree with numerical simulations. Finally, we analyse the consequences of non-homogeneous surface charge density. We conclude that the geometry of cusp-shaped domains influences the voltage profile, specifically inside the cusp structure. The main results are summarized in the form of new three-dimensional electrostatic laws for non-electroneutral electrolytes. We discuss applications to dendritic spines in neuroscience., Comment: 28 pages, 6 figures

To recover the long-time behavior and the statistics of molecular trajectories from the large number (tens of thousands) of their short fragments, obtained by super-resolution methods at the single molecule level, data analysis based on a stochastic model of their non-equilibrium motion is required. Recently, we characterized the local biophysical properties underlying receptor motion based on coarse-grained long-range interactions, corresponding to attracting potential wells of large sizes. The purpose of this letter is to discuss optimal estimators and show what happens when thing goes wrong., Comment: 4 pages

The Poisson Nernst-Planck equations for charge concentration and electric potential in a ball is a model of electro-diffusion of ions in the head of a neuronal dendritic spine. We study the relaxation and the steady state when an initial charge of ions is injected into the ball. The steady state equation is similar to the Liouville-Gelfand-Bratu-type equation with the difference that the boundary condition is Neumann, not Dirichlet and there a minus sign in the exponent of the exponential term. The entire boundary is impermeable to the ions and the electric field satisfies the compatibility condition of Poisson's equation. We construct a steady radial solution and find that the potential is maximal in the center and decreases toward the boundary. We study the limit of large charge in dimension 1,2 and 3. For the case of a small absorbing window in the sphere, we find the escape rate of an ion from the steady density., Comment: 20 pages in SIAP 2015

A key limiting step in fertility is the search for the oocyte by spermatozoa. Initially, there are tens of millions of sperm cells, but a single one will make it to the oocyte. This may be one of the most severe selection processes designed by evolution, whose role is yet to be understood. Why is it that such a huge redundancy is required and what does that mean for the search process? we propose to discuss here these questions and consequently a new line of interdisciplinary research needed to find possible answers., Comment: 6 pages

We study the escape of Brownian motion from the domain of attraction $\Omega$ of a stable focus with a strong drift. The boundary $\partial \Omega$ of $\Omega$ is an unstable limit cycle of the drift and the focus is very close to the limit cycle. We find a new phenomenon of oscillatory decay of the peaks of the survival probability of the Brownian motion in $\Omega$. We compute explicitly the complex-valued second eigenvalue $\lambda_2(\Omega$) of the Fokker-Planck operator with Dirichlet boundary conditions and show that it is responsible for the peaks. Specifically, we demonstrate that the dominant oscillation frequency equals $1/{\mathfrak{I}}m\{\lambda_2(\Omega)\}$ and is independent of the relative noise strength. We apply the analysis to a canonical system and compare the density of exit points on $\partial \Omega$ to that obtained from stochastic simulations. We find that this density is concentrated in a small portion of the boundary, thus rendering the exit a narrow escape problem. Unlike the case in the classical activated escape problem, the principal eigenvalue does not necessarily decay exponentially as the relative noise strength decays. The oscillatory narrow escape problem arises in a mathematical model of neural networks with synaptic depression. We identify oscillation peaks in the density of the time the network spends in a specific state. This observation explains the oscillations of stochastic trajectories around a focus prior to escape and also the non-Poissonian escape times. This phenomenon has been observed and reported in the neural network literature., Comment: 25 p

By combining high-density super-resolution imaging with a novel stochastic analysis, we report here a peculiar nano-structure organization revealed by the density function of individual AMPA receptors moving on the surface of cultured hippocampal dendrites. High density regions of hundreds of nanometers for the trajectories are associated with local molecular assembly generated by direct molecular interactions due to physical potential wells. We found here that for some of these regions, the potential wells are organized in ring structures. We could find up to 3 wells in a single ring. Inside a ring receptors move in a small band the width of which is of hundreds of nanometers. In addition, rings are transient structures and can be observed for tens of minutes. Potential wells located in a ring are also transient and the position of their peaks can shift with time. We conclude that these rings can trap receptors in a unique geometrical structure contributing to shape receptor trafficking, a process that sustains synaptic transmission and plasticity., Comment: 4 figures extension of Hoze et al, PNAS 2012

Intracellular transport of DNA carriers is a fundamental step of gene delivery. We present here a theoretical approach to study generically a single virus or DNA particle trafficking in a cell cytoplasm. Cellular trafficking has been studied experimentally mostly at the macroscopic level, but very little has been done so far at the microscopic level. We present here a physical model to account for certain aspects of cellular organization, starting with the observation that a viral particle trajectory consists of epochs of pure diffusion and epochs of active transport along microtubules. We define a general degradation rate to describe the limitations of the delivery of plasmid or viral particles to the nucleus imposed by various types of direct and indirect hydrolysis activity inside the cytoplasm. Following a homogenization procedure, which consists of replacing the switching dynamics by a single steady state stochastic description, not only can we study the spatio-temporal dynamics of moving objects in the cytosol, but also estimate the probability and the mean time to go from the cell membrane to a nuclear pore. Computational simulations confirm that our model can be used to analyze and interpret viral trajectories and estimate quantitatively the success of nuclear delivery., Comment: 10 pages, 6 figures

The radiation (reaction, Robin) boundary condition for the continuum diffusion equation is widely used in chemical and biological applications to express reactive boundaries. The underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary, however, the relation between the reaction (radiation) constant in the Robin boundary condition and the reflection probability is still unclear. In this paper we clarify the issue by finding the relation between the reaction (radiation) constant and the absorption probability of the diffusing trajectories at the boundary. We analyze the Euler scheme for the underlying It\^o dynamics, which is assumed to have variable drift and diffusion tensor, with partial reflection at the boundary. Trajectories that cross the boundary are terminated with a given probability and otherwise are reflected in a normal or oblique direction. We use boundary layer analysis of the corresponding Wiener path integral to resolve the non-uniform convergence of the probability density function of the numerical scheme to the solution of the Fokker-Planck equation with the Robin boundary condition, as the time step is decreased. We show that the Robin boundary condition is recovered in the limit iff trajectories are reflected in the co-normal direction. We find the relation of the reactive constant to the termination probability. We show the effect of using the new relation in numerical simulations.

We present a general framework to study the effect of killing sources on moving particles, trafficking inside biological cells. We are merely concerned with the case of spine-dendrite communication, where the number of calcium ions, modeled as random particles is regulated across the spine microstructure by pumps, which play the killing role. In particular, we study here the survival probability of ions in such environment and we present a general theory to compute the ratio of the number of absorbed particles at specific location to the number of killed particles during their sojourn inside a domain. In the case of a dendritic spine, the ratio is computed in terms of the survival probability of a stochastic trajectory in a one dimensional approximation. We show that the ratio depends on the distribution of killing sources. The biological conclusion follows: changing the position of the pumps is enough to regulate the calcium ions and thus the spine-dendrite communication.