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We develop a computational approach to locate the source of a steady-state gradient of diffusing particles from the fluxes through narrow windows distributed either on the boundary of a three dimensional half-space or on a sphere. This approach is based on solving the mixed boundary stationary diffusion equation with the Neumann-Green's function and matched asymptotic. We compute the probability fluxes and develop a highly efficient analytical-Brownian numerical scheme. This scheme accelerates the simulation time by avoiding the explicit computation of Brownian trajectories in the infinite domain. Our derived analytical formulas agree with the results obtained from the fast numerical simulation scheme. Using the analytical representation of the particle fluxes, we show how to reconstruct the location of the point source. Furthermore, we investigate the uncertainty in the source reconstruction due to additive fluctuations present in the fluxes. We also study the influence of various window configurations (cluster vs uniform distributions) on recovering the source position.

We study here the extreme statistics of Brownian particles escaping from a cusp funnel: the fastest Brownian particles among $n$ follow an ensemble of optimal trajectories located near the shortest path from the source to the target. For the time of such first arrivers, we derive an asymptotic formula that differs from the classical narrow escape and dire strait obtained for the mean first passage time. Consequently, when particles are initially distributed at a given distance from a cusp, the fastest do see some properties characterizing the cusp geometry. Therefore, when many particles diffuse around impermeable obstacles, the geometry plays a role in the time to reach a target. In the biological context of cellular transduction with signalling molecules, having to escape such cusp-like domains slows down fast signaling. To conclude, generating multiple copies of the same molecule helps bypass a crowded environment to transmit a molecular signal quickly., Comment: 5p

The current-voltage (I-V) conversion characterizes the physiology of cellular microdomains and reflects cellular communication, excitability, and electrical transduction. Yet deriving such I-V laws remains a major challenge in most cellular microdomains due to their small sizes and the difficulty of accessing voltage with a high nanometer precision. We present here novel analytical relations derived for different numbers of ionic species inside a neuronal micro/nano-domains, such as dendritic spines. When a steady-state current is injected, we find a large deviation from the classical Ohm's law, showing that the spine neck resistance is insuficent to characterize electrical properties. For a constricted spine neck, modeled by a hyperboloid, we obtain a new I-V law that illustrates the consequences of narrow passages on electrical conduction. Finally, during a fast current transient, the local voltage is modulated by the distance between activated voltage-gated channels. To conclude, electro-diffusion laws can now be used to interpret voltage distribution in neuronal microdomains., Comment: 3 figures;submitted to PRL

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

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

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

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

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

Analysis of high-density superresolution imaging of receptors reveal the organization of dendrites at the nano-scale resolution. We present here simulations in empirical live cell images, which allows converting local information extracted from short range trajectories into simulations of long range trajectories. Based on these empirical simulations, we compute the residence time of an AMPA receptor (AMPAR) in dendritic spines that accounts for receptors local interactions and geometrical organization. We report here that depending on the type of the spine, the residence time varies from one to five minutes. Moreover, we show that there exists transient organized structures, previously described as potential wells that can regulate the trafficking of AMPARs to dendritic spines., Comment: 19 pages

Activated escape of a Brownian particle from the domain of attraction of a stable focus over a limit cycle exhibits non-Kramers behavior: it is non-Poissonian. When the attractor is moved closer to the boundary oscillations can be discerned in the survival probability. We show that these oscillations are due to complex-valued higher order eigenvalues of the Fokker-Planck operator, which we compute explicitly in the limit of small noise. We also show that in this limit the period of the oscillations is the winding number of the activated stochastic process. These peak probability oscillations are not related to stochastic resonance and should be detectable in planar dynamical systems with the topology described here., Comment: Oscillation peaks. Asymptotic computation of the spectrum for a non-selfadjoint second order operator

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

We study the first passage time for a polymer, that we call the narrow encounter time (NETP), to reach a small target located on the surface of a microdomain. The polymer is modeled as a Freely Joint Chain (beads connected by springs with a resting non zero length) and we use Brownian simulations to study two cases: when (i) any of the monomer or (ii) only one can be absorbed at the target window. Interestingly, we find that {in the first case} the NETP is an increasing function of the polymer length until a critical length, after which it decreases. Moreover, in the long polymer regime, we identified an exponential scaling law for the NETP as a function of the polymer length. {In the second case, the position of the absorbed monomer along the polymer chain strongly influences the NETP}. Our analysis can be applied to estimate the mean first time of a DNA fragment to a small target in the chromatin structure or for mRNA to find a small target., Comment: LaTeX, 25 pages, 6 eps figures

We study the probability of two Brownian particles to meet before one of them exits a finite interval. We obtain an explicit expression for the probability as a function of the initial distance of the two particles using the Weierstrass elliptic function. We also find the law of the meeting location. Brownian simulations show the accuracy of our analysis. Finally, we discuss some applications to the probability that a double strand DNA break repairs in confined environments., Comment: To appear J. Phys A

Questions of flux regulation in biological cells raise renewed interest in the narrow escape problem. The often inadequate expansions of the narrow escape time are due to a not so well known fact that the boundary singularity of Green's function for Poisson's equation with Neumann and mixed Dirichlet-Neumann boundary conditions in three-dimensions contains a logarithmic singularity. Using this fact, we find the second term in the expansion of the narrow escape time and in the expansion of the principal eigenvalue of the Laplace equation with mixed Dirichlet-Neumann boundary conditions, with small Dirichlet and large Neumann parts. We also find the leakage flux of Brownian particles that diffuse from a source to an absorbing target on a reflecting boundary of a domain, if a small perforation is made in the reflecting boundary.

Active cellular transport is a fundamental mechanism for protein and vesicle delivery, cell cycle and molecular degradation. Viruses can hijack the transport system and use it to reach the nucleus. Most transport processes consist of intermittent dynamics, where the motion of a particle, such as a virus, alternates between pure Brownian and directed movement along microtubules. In this communication, we estimate the mean time for particle to attach to a microtubule network. This computation leads to a coarse grained equation of the intermittent motion in radial and cylindrical geometries. Finally, by using the degradation activity inside the cytoplasm, we obtain refined asymptotic estimations for the probability and the mean time a virus reaches a small nuclear pore., Comment: 4 pages, 5 figures accepted as rapid communication in Phys. Rev. E

The mean time required by a transcription factor (TF) or an enzyme to find a target in the nucleus is of prime importance for the initialization of transcription, gene activation or the start of DNA repair. We obtain new estimates for the mean search time when the TF or enzyme, confined to the cell nucleus, can switch from a one dimensional motion along the DNA and a free Brownian regime inside the crowded nucleus. We give analytical expressions for the mean time the particle stays bound to the DNA, $\tau_{DNA}$, and the mean time it diffuses freely, $\tau_{free}$. Contrary to previous results but in agreement with experimental data, we find a factor $\tau_{DNA} \approx 3.7 \tau_{free}$ for the Lac-I TF. The formula obtained for the time required to bind to a target site is found to be coherent with observed data. We also conclude that a higher DNA density leads to a more efficient search process.

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 study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold when the diffusion constant $\epsilon$ goes to zero. We assume that the first order term is given by a vector field $b$, whose recurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the normalized eigenfunctions concentrate on the recurrent sets of maximal dimension where the topological pressure \cite{Kifer90} is attained. On the cycles and torii, the limit measures are absolutely continuous with respect to the invariant probability measure on these sets. We have determined these limit measures, using a blow-up analysis., Comment: Note to appear in C.R.A.S

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.

In 1941 Sumner Myers proved that if the Ricci curvature of a complete Riemann manifold has a positive infimum then the manifold is compact and its diameter is bounded in terms of the infimum. Subsequently the curvature hypothesis has been weakened, and in this paper we weaken it further in an attempt to find the ultimate, sharp result., Comment: 24 pages,3 figures, good stuff. submitted

Traditional chemical kinetics may be inappropriate to describe chemical reactions in micro-domains involving only a small number of substrate and reactant molecules. Starting with the stochastic dynamics of the molecules, we derive a master-diffusion equation for the joint probability density of a mobile reactant and the number of bound substrate in a confined domain. We use the equation to calculate the fluctuations in the number of bound substrate molecules as a function of initial reactant distribution. A second model is presented based on a Markov description of the binding and unbinding and on the mean first passage time of a molecule to a small portion of the boundary. These models can be used for the description of noise due to gating of ionic channels by random binding and unbinding of ligands in biological sensor cells, such as olfactory cilia, photo-receptors, hair cells in the cochlea., Comment: 33 pages, Journal Chemical Physics

We consider Brownian motion in a bounded domain $\Omega$ on a two-dimensional Riemannian manifold $(\Sigma,g)$. We assume that the boundary $\p\Omega$ is smooth and reflects the trajectories, except for a small absorbing arc $\p\Omega_a\subset\p\Omega$. As $\p\Omega_a$ is shrunk to zero the expected time to absorption in $\p\Omega_a$ becomes infinite. The narrow escape problem consists in constructing an asymptotic expansion of the expected lifetime, denoted $E\tau$, as $\epsilon=|\partial \Omega_a|_g/|\partial \Omega|_g\to0$. We derive a leading order asymptotic approximation $E\tau = \ds{\frac{|\Omega|_g}{D\pi}}[\log\ds{\frac{1}{\epsilon}}+O(1)]$. The order 1 term can be evaluated for simply connected domains on a sphere by projecting stereographically on the complex plane and mapping conformally on a circular disk. It can also be evaluated for domains that can be mapped conformally onto an annulus. This term is needed in real life applications, such as trafficking of receptors on neuronal spines, because $\log\ds{\frac{1}{\epsilon}}$ is not necessarily large, even when $\epsilon$ is small. If the absorbing window is located at a corner of angle $\alpha$, then $E\tau = \ds{\frac{|\Omega|_g}{D\alpha}}[\log\ds{\frac{1}{\epsilon}}+O(1)],$ if near a cusp, then $E\tau$ grows algebraically, rather than logarithmically. Thus, in the domain bounded between two tangent circles, the expected lifetime is $E\tau = \ds{\frac{|\Omega|}{(d^{-1}-1)D}}(\frac{1}{\epsilon} + O(1))$., Comment: This is the third in a series of three papers

A Brownian particle with diffusion coefficient $D$ is confined to a bounded domain of volume $V$ in $\rR^3$ by a reflecting boundary, except for a small absorbing window. The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. We construct an asymptotic approximation for the case of an elliptical window of large semi axis $a\ll V^{1/3}$ and show that the mean escape time is $E\tau\sim\ds{\frac{V}{2\pi Da}} K(e)$, where $e$ is the eccentricity of the ellipse; and $K(\cdot)$ is the complete elliptic integral of the first kind. In the special case of a circular hole the result reduces to Lord Rayleigh's formula $E\tau\sim\ds{\frac{V}{4aD}}$, which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion $E\tau=\ds{\frac{V}{4aD}} [1+\frac{a}{R} \log \frac{R}{a} + O(\frac{a}{R}) ]$. This problem is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function., Comment: This is the first in a series of three papers

We consider Brownian motion in a circular disk $\Omega$, whose boundary $\p\Omega$ is reflecting, except for a small arc, $\p\Omega_a$, which is absorbing. As $\epsilon=|\partial \Omega_a|/|\partial \Omega|$ decreases to zero the mean time to absorption in $\p\Omega_a$, denoted $E\tau$, becomes infinite. The narrow escape problem is to find an asymptotic expansion of $E\tau$ for $\epsilon\ll1$. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results improve the previously derived expansion for a general smooth domain, $E\tau = \ds{\frac{|\Omega|}{D\pi}}[\log\ds{\frac{1}{\epsilon}}+O(1)],$ ($D$ is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of the disk is $E[\tau | \x(0)=\mb{0}]=\ds{\frac{R^2}{D}}[\log\ds{\frac{1}{\epsilon}} + \log 2 +\ds{{1/4}} + O(\epsilon)]$. The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because $\log\ds{\frac{1}{\epsilon}}$ is not necessarily large, even when $\epsilon$ is small. We also find the singular behavior of the probability flux profile into $\p\Omega_a$ at the endpoints of $\p\Omega_a$, and find the value of the flux near the center of the window., Comment: This is the second in a series of three papers

In this paper, we give explicit estimates that insure the existence of solutions for first order partial differential operators on compact manifolds, using a viscosity method. In the linear case, an explicit integral formula can be found, using the characteristics curves. The solution is given explicitly on the critical points and the limit cycles of the vector field of the first order term of the operator. In the nonlinear case, a generalization of the Weitzenbock formula provides pointwise estimates that insure the existence of a solution, but the uniqueness question is left open. Nevertheless we prove that uniqueness is stable under a C^{1} perturbation. Finally, we give some examples where the solution fails to exist globally, justifying the need to impose conditions that warrant global existence. The last result reveals that the zero order term in the first order operator is necessary to obtain generically bounded solutions., Comment: 32 pages, to appear in Quarterly Journal of Mathematics

In this note we present some results concerning the concentration of sequences of first eigenfunctions on the limit sets of a Morse-Smale dynamical system on a compact Riemanniann manifold. More precisely a renormalized sequence of eigenfunctions converges to a measure $\mu$ concentrated on the hyperbolic sets of the field. The set of all possible measure turns out to be a sum of a finite Dirac distributions localized at the critical point of the field and absolutely continuous measure with respect to the Lebesgue measure on each limit cycles : the coefficients which appear in the limit measure can be characterized using the concentration theory. In the second part, certain aspects of some first order PDE on manifolds are studied. We study the limit of a sequence solutions of a second order PDE, when a parameter of viscosity tends to zero. Under some explicit assumptions on some vector fields, bounded and differentiable solutions are obtained. We exibit the role played by the limit sets of the dynamical systems and provide in some cases an explicit representation formula., Comment: 6 pages, Latex