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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 mean first passage times obtained for classical narrow escape and dire strait. When particles are initially distributed at a given distance from a cusp, the time of the fastest particles depends on the cusp geometry. Therefore, when many particles diffuse around impermeable obstacles, the geometry plays a role in the time it takes to reach a target. In the context of cellular transduction with signaling molecules, having to escape from such cusp-like domains slows down signaling pathways. Consequently, generating multiple copies of the same molecule enables molecular signals to be delivered through crowded environments in sufficient time. [ABSTRACT FROM AUTHOR]

The first arrivals among N Brownian particles is ubiquitous in the life sciences, as it often triggers cellular processes from the molecular level. We study here the case where stochastic particles, which represent proteins or molecules can switch between two states inside the non-negative real line. The switching process is modeled as a two-state Markov chain and particles can only escape in state 1. We estimate the fastest arrival time by solving asymptotically the Fokker–Planck equations for three different initial distributions: Dirac-delta, uniformly distributed and long-tail decay. The derived formulas reveal that the fastest particle avoids switching when the switching rates are much smaller than the diffusion time scale, but switches twice when the diffusion in state 2 is much faster than in state 1. The present results are compared to stochastic simulations revealing the range of validity of the derived formulas. [ABSTRACT FROM AUTHOR]

When a flux of Brownian particles is injected in a narrow window located on the surface of a bounded domain, these particles diffuse and can eventually escape through a cluster of narrow windows. At steady state, we compute asymptotically the distribution of concentration between the different windows. The solution is obtained by solving Laplace's equation using Green's function techniques and second-order asymptotic analysis, and depends on the influx amplitude, the diffusion properties, as well as the geometrical organization of all the windows, such as their distances and the mean curvature. We explore the range of validity of the present asymptotic expansions using numerical simulations of the mixed boundary value problem. Finally, we introduce a length scale to estimate how deep inside a domain a local diffusion current can spread. [ABSTRACT FROM AUTHOR]

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 Neumann–Green's function. The method of matched asymptotic expansions enables the computation of the probability fluxes. To explore the range of validity of this expansion, we develop a fast analytical-Brownian numerical scheme. This scheme accelerates the simulation time by avoiding the explicit computation of Brownian trajectories in the infinite domain. The results obtained from our derived analytical formulae and the fast numerical simulation scheme agree on a large range of parameters. 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: clustered versus uniform distributions on recovering the source position. Finally, we discuss possible applications for cell navigation in biology. [ABSTRACT FROM AUTHOR]

Biological networks are characterized by their connectivity and topology but also by their ability to transport materials. In the case of random transportation, the efficacy is measured by the time it takes to travel between two nodes of the network. We study here the consequences of a unidirectional transport mechanism occurring in the endoplasmic reticulum (ER) network, a structure present in the cell cytoplasm. This unidirectional transport mechanism is an active-waiting transportation, where molecules have to wait a random time before being transported from one node to the next one. We develop here a general theory of transport in an active network and find an unusual network transportation, where molecules group together in redundant packets instead of being disperse. Finally, the mean time to travel between two nodes of the ER is of the order of 20 min, but is reduced to 30 s when we consider the fastest particles because it uses optimal paths. To conclude, the present theory shows that unidirectional transport is an efficient and robust mechanism for fast molecular redistribution inside the ER. [ABSTRACT FROM AUTHOR]

Traditional chemical kinetics may be inappropriate to describe chemical reactions in microdomains 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, photoreceptors, hair cells in the cochlea. [ABSTRACT FROM AUTHOR]

We study here the bulk electro-diffusion properties of micro- and nanodomains containing a cusp-shaped structure in three-dimensions when the cation concentration dominates over the anions. To determine the consequences on the voltage distribution, we use the steady-state Poisson–Nernst–Planck 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 surface charge distribution that agree with numerical simulations. Finally, we analyze the consequences of several piecewise constant non-homogeneous surface charge densities, motivated by designing new nanopipettes. To conclude, when electro-neutrality is broken at the scale of hundreds of nanometers, 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. These formula provide a refined characterization of voltage distribution at steady-state in neuronal microdomains such as dendritic spines, but can also be used to design nanometric patch-pipettes. [ABSTRACT FROM AUTHOR]

Is it possible to recover the position of a source from the steady-state fluxes of Brownian particles to small absorbing windows located on the boundary of a domain? To address this question, we develop a numerical procedure to avoid tracking Brownian trajectories in the entire infinite space. Instead, we generate particles near the absorbing windows, computed from the analytical expression of the exit probability. When the Brownian particles are generated by a steady-state gradient at a single point, we compute asymptotically the fluxes to small absorbing holes distributed on the boundary of half-space and on a disk in two dimensions, which agree with stochastic simulations. We also derive an expression for the splitting probability between small windows using the matched asymptotic method. Finally, when there are more than two small absorbing windows, we show how to reconstruct the position of the source from the diffusion fluxes. The present approach provides a computational first principle for the mechanism of sensing a gradient of diffusing particles, a ubiquitous problem in cell biology. [ABSTRACT FROM AUTHOR]

The main difficulty in simulating diffusion processes at a molecular level in cell microdomains is due to the multiple scales involving nano- to micrometers. Few to many particles have to be simulated and simultaneously tracked while there are exploring a large portion of the space for binding small targets, such as buffers or active sites. Bridging the small and large spatial scales is achieved by rare events representing Brownian particles finding small targets and characterized by long-time distribution. These rare events are the bottleneck of numerical simulations. A naive stochastic simulation requires running many Brownian particles together, which is computationally greedy and inefficient. Solving the associated partial differential equations is also difficult due to the time dependent boundary conditions, narrow passages and mixed boundary conditions at small windows. We present here two reduced modeling approaches for a fast computation of diffusing fluxes in microdomains. The first approach is based on a Markov mass-action law equations coupled to a Markov chain. The second is a Gillespie's method based on the narrow escape theory for coarse-graining the geometry of the domain into Poissonian rates. The main application concerns diffusion in cellular biology, where we compute as an example the distribution of arrival times of calcium ions to small hidden targets to trigger vesicular release. [ABSTRACT FROM AUTHOR]

We review here recent progress to link the nuclear organization to its function, based on elementary physical processes such as diffusion, polymer dynamics of DNA, chromatin and the search mechanism for a small target by double-stranded DNA (dsDNA) break. These physical models and their analysis make it possible to compute critical rates involved in cell reorganization timing, which depend on many parameters. In the framework of polymer models, various empirical observations are interpreted as anomalous diffusion of chromatin at various time scales. The reviewed theoretical approaches offer a framework for extracting features, biophysical parameters, predictions, and so on, based on a large variety of experimental data, such as chromosomal capture data, single particle trajectories, and more. Combining theoretical approaches with live cell microscopy data should unveil some of the still unexplained behavior of the nucleus in carrying out some of its key function involved in survival, DNA repair or gene activation. [ABSTRACT FROM AUTHOR]

Many viruses must enter the cell nucleus through small nanopores in order to replicate. We model here the viral motion as a stochastic process described by the Survival Fokker-Planck equation. We estimate the probability and the conditional mean first passage time that a viral trajectory is absorbed at a small nuclear pore before being terminated. The method is based on the explicit Neumann-Green's function. The cell nucleus is modeled as a three dimensional ball, covered with thousands of small absorbing windows. The minimum distance between them defines the smallest spatial scale that is an unavoidable limit for efficient stochastic simulations. Derived asymptotic formula agree with stochastic simulations and reveal how small and large geometrical parameters define the cytoplasmic stage of viral infection. [ABSTRACT FROM AUTHOR]

Recovering a stochastic process from noisy ensembles of single-particle trajectories is resolved here using the coarse-grained Langevin equation as a model. The massive redundancy contained in single-particle tracking data allows recovering local parameters of the underlying physical model. We use several parametric and nonparametric estimators to compute the first and second moments of the process, to recover the local drift, its derivative, and the diffusion tensor, and to deconvolve the instrumental from the physical noise. We use numerical simulations to also explore the range of validity for these estimators. The present analysis allows defining what can exactly be recovered from statistics of super-resolution microscopy trajectories used for characterizing molecular trafficking underlying cellular functions. [ABSTRACT FROM AUTHOR]

The arrival of a Brownian particle at a narrow cusp located underneath a ball is a model of vesicular release at neuronal synapses, triggered by calcium ions. The asymptotic computation of the arrival time presents several difficulties that can be overcome using conformal mappings and asymptotic analysis of the model equations. Using a regular expansion of the solution of the Laplace equation in the mapped domain, we compute the solution involving both small and large spatial scales. We derive novel asymptotic formulas for Brownian escape through cusps in both two and three dimensions. The range of validity of the asymptotic formulas is challenged by stochastic simulations. Finally, we apply the analysis to estimate the vesicular release probability at presynaptic terminals and, in particular, we suggest that vesicular organization imposes a severe constraint on calcium channel localization: diffusing calcium ions can trigger vesicular release only in a specific range of positions that we provide. [ABSTRACT FROM AUTHOR]

Messenger RNAs (mRNAs) can be repressed and degraded by small non-coding RNA molecules. In this paper, we formulate a coarsegrained Markov-chain description of the post-transcriptional regulation of mRNAs by either small interfering RNAs (siRNAs) or microRNAs (miRNAs). We calculate the probability of an mRNA escaping from its domain before it is repressed by siRNAs/miRNAs via calculation of the mean time to threshold: when the number of bound siRNAs/miRNAs exceeds a certain threshold value, the mRNA is irreversibly repressed. In some cases, the analysis can be reduced to counting certain paths in a reduced Markov model. We obtain explicit expressions when the small RNA bind irreversibly to the mRNA and we also discuss the reversible binding case. We apply our models to the study of RNA interference in the nucleus, examining the probability of mRNAs escaping via small nuclear pores before being degraded by siRNAs. Using the same modelling framework, we further investigate the effect of small, decoy RNAs (decoys) on the process of post-transcriptional regulation, by studying regulation of the tumor suppressor gene, PTEN: decoys are able to block binding sites on PTEN mRNAs, thereby reducing the number of sites available to siRNAs/miRNAs and helping to protect it from repression. We calculate the probability of a cytoplasmic PTEN mRNA translocating to the endoplasmic reticulum before being repressed by miRNAs. We support our results with stochastic simulations. [ABSTRACT FROM AUTHOR]

We demonstrate the oscillatory decay of the survival probability of the stochastic dynamics dxε = a(xε) dt+ √ 2ε b(xε) dw, which is activated by small noise over the boundary of the domain of attraction D of a stable focus of the drift a(x). The boundary ∂D of the domain is an unstable limit cycle of a(x). The oscillations are explained by a singular perturbation expansion of the spectrum of the Dirichlet problem for the non-self-adjoint Fokker-Planck operator inD Lεu(x) = εΣ2i,j=1 ∂2[σi,j(x)u(x)]/∂xi∂xj -Σ2i=1 ∂[ai(x)u(x)]/ ∂xi = -λεu(x), with σ(x) = b(x)bT (x). We calculate the leading-order asymptotic expansion of all eigenvalues λε for small ε. The principal eigenvalue is known to decay exponentially fast as ε → 0. We find that for small ε the higher-order eigenvalues are given by λm,n = nω1+miω2+O(ε) for n = 1, 2, . . . , m = ±1, . . . , where ω1 and ω2 are explicitly computed constants. We also find the asymptotic structure of the eigenfunctions of Lε and of its adjoint L*ε. We illustrate the oscillatory decay with a model of synaptic depression of neuronal networks in neurobiology. [ABSTRACT FROM AUTHOR]

Diffusion is the driver of critical biological processes in cellular and molecular biology. The diverse temporal scales of cellular function are determined by vastly diverse spatial scales in most biophysical processes. The latter are due, among others, to small binding sites inside or on the cell membrane or to narrow passages between large cellular compartments. The great disparity in scales is at the root of the difficulty in quantifying cell function from molecular dynamics and from simulations. The coarse-grained time scale of cellular function is determined from molecular diffusion by the mean first passage time of molecular Brownian motion to a small targets or through narrow passages. The narrow escape theory (NET) concerns this issue. The NET is ubiquitous in molecular and cellular biology and is manifested, among others, in chemical reactions, in the calculation of the effective diffusion coefficient of receptors diffusing on a neuronal cell membrane strewn with obstacles, in the quantification of the early steps of viral trafficking, in the regulation of diffusion between the mother and daughter cells during cell division, and many other cases. Brownian trajectories can represent the motion of a molecule, a protein, an ion in solution, a receptor in a cell or on its membrane, and many other biochemical processes. The small target can represent a binding site or an ionic channel, a hidden active site embedded in a complex protein structure, a receptor for a neurotransmitter on the membrane of a neuron, and so on. The mean time to attach to a receptor or activator determines diffusion fluxes that are key regulators of cell function. This review describes physical models of various subcellular microdomains, in which the NET coarse-grains the molecular scale to a higher cellular-level, thus clarifying the role of cell geometry in determining subcellular function. [ABSTRACT FROM AUTHOR]

The narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain. The problem is equivalent to solving the mixed Dirichlet--Neumann boundary value problem for the Poisson equation with small Dirichlet and large Neumann parts. The mixed boundary value problem, which goes back to Lord Rayleigh, originates in the theory of sound and is closely connected to the eigenvalue problem for the mixed problem and for the Neumann problem in domains with bottlenecks. We review here recent developments in the non- standard asymptotics of the problem, which are based on several ingredients: a better resolution of the singularity of Neumann's function, resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components. The new methodology applies to two- and higher-dimensional problems. Selected applications are reviewed. [ABSTRACT FROM AUTHOR]

The telomere length can either be shortened or elongated by an enzyme called telomerase after each cell division. Interestingly, the shortest telomere is involved in controlling the ability of a cell to divide. Yet, its dynamics remains elusive. We present here a stochastic approach where we model this dynamics using a Markov jump process. We solve the forward Fokker-Planck equation to obtain the steady state distribution and the statistical moments of telomere lengths. We focus specifically on the shortest one and we estimate its length difference with the second shortest telomere. After extracting key parameters such as elongation and shortening dynamics from experimental data, we compute the length of telomeres in yeast and obtain as a possible prediction the minimum concentration of telomerase required to ensure a proper cell division. [ABSTRACT FROM AUTHOR]

Chromatin inside the cell nucleus consists of the DNA and its hierarchy of interacting molecules, which can be modeled as a complex polymer. To describe the chromatin dynamic, we develop and analyze here a polymer model that accounts for long-range interactions and not just those between the closest neighbors as in the Rouse polymer model. Our construction of the polymer model allows us to recover the local interaction between monomers from the anomalous diffusion exponent, which can be directly measured experimentally. We compute asymptotically for this polymer model the cross-correlation function for a given monomer and the mean time for a loop to be formed. Finally, we discuss some possible applications for interpretation of chromosome capture data. [ABSTRACT FROM AUTHOR]

Critical biological processes, such as synaptic plasticity and transmission, activation of genes by transcription factors, or double-strained DNA break repair, are controlled by diffusion in structures that have both large and small spatial scales. These may be small binding sites inside or on the surface of the cell, or narrow passages between subcellular compartments. The great disparity in spatial scales is the key to controlling cell function by structure. We report here recent progress on resolving analytical and numerical difficulties in extracting properties from experimental data, from biophysical models, and from Brownian dynamics simulations of diffusion in multi-scale structures. This progress is achieved by developing an analytical approximation methodology for solving the model equations. The reported results are applied to analysis and simulations of subcellular processes and to the quantification of their biological functions. [ABSTRACT FROM AUTHOR]

Is it possible to extract the size and structure of chromosomal territories (confined domain) from the encounter frequencies of chromosomal loci? To answer this question, we estimate the mean time for two monomers located on the same polymer to encounter, which we call the mean first encounter time in a confined microdomain (MFETC). We approximate the confined domain geometry by a harmonic potential well and obtain an asymptotic expression that agrees with Brownian simulations for the MFETC as a function of the polymer length, the radius of the confined domain, and the activation distance radius ε at which the two searching monomers meet. We illustrate the present approach using chromosome capture data for the encounter rate distribution of two loci depending on their distances along the DNA. We estimate the domain size that restricts the motion of one of these loci for chromosome II in yeast. [ABSTRACT FROM AUTHOR]

The passage of Brownian motion through a bottleneck in a bounded domain is a rare event, and as the bottleneck radius shrinks to zero the mean time for such passage increases indefinitely. Its calculation reveals the effect of geometry and smoothness on the flux through the bottleneck. We find new behavior of the narrow escape time through bottlenecks in planar and spatial domains and on a surface. Some applications in cellular biology and neurobiology are discussed. [ABSTRACT FROM AUTHOR]

We study the mean turnaround time of a Brownian needle in a narrow planar strip. When the needle is only slightly shorter than the width of the strip, the computation becomes a nonstandard narrow escape problem. We develop a boundary layer method, based on a conformal mapping of cusplike narrow straits, to obtain an explicit asymptotic approximation to the mean turnaround time. Our result suggests that two-dimensional domains lying between parallel walls may play a significant role in DNA repair. [ABSTRACT FROM AUTHOR]

Particles diffusing on a membrane crowded with obstacles have to squeeze between them through funnel-shaped narrow straits. The computation of the mean passage time through the straits is a new narrow escape problem that gives rise to new, hitherto unknown, behavior that we communicate here. The motion through the straits on the coarse scale of the narrow escape time is an effective diffusion with coefficient that varies nonlinearly with the density of obstacles. We calculate the coarse-grained diffusion coefficient on a planar lattice of circular obstacles and use it to estimate the density of obstacles on a neuronal membrane and in a model of a cytoplasm crowded by identical parallel circular rods. [ABSTRACT FROM AUTHOR]

Abstract: The mean time for a transcription factor (TF) or a DNA-ligation enzyme, to find their target located on the molecule DNA is crucial for cell activation or survival. Using mean first passage time equations, we obtain here refined estimates for , based on the classical scenario in which the search process switches between a one-dimensional motion along the DNA molecule and a free Brownian motion in the nucleus. By considering the interaction potential between the TF and the DNA base pairs (bp), we find that the TF stays attached to the DNA molecule during an average time . Between two consecutive attached periods, the TF diffuses freely inside the nucleus during . In agreement with experimental data for the Lac-I TF, we find that . [Copyright &y& Elsevier]

Abstract: Morphogens are molecules inducing morphogenetic responses from cells and cell ensembles. The concept of morphogen is related to that of positional value, as the generation of morphological and physiological characteristics is function of position. Based on the observation that homeoproteins, a category of transcription factors with morphogenetic functions, traffic between abutting cells and, very often, regulate their own expression, we develop here a biophysical model of homeoprotein propagation and study the associated mathematical equations. This mode of cell signaling can generate domains of homeoprotein expression. We study both the transient and steady-state regimes and, in this latter regime, we obtain various morphogenetic gradients, depending on the value of some parameters, such as morphogen synthesis, degradation rates and efficiency of intercellular passage. The same equations, applied to pairs of homeoproteins with auto-activation and reciprocal inhibition properties, account for border formation. They also allow us to compute how specific perturbations can either be buffered or lead to modifications in the position of borders between adjacent areas. The model developed here, based on experimental data, and avoids theoretical obstacles associated with pluricellularity. It extends the idea that Bicoid homeoprotein is a morphogen in the fly embryo syncitium to most homeoproteins and to pluricellular systems. Because the position of borders between brain areas is of primary physiological importance, our model might lead to original views regarding epigenetic inter-individual variations and the origin of neurological and psychiatric diseases. In addition, it provides new hypotheses regarding the molecular basis of brain evolution. [Copyright &y& Elsevier]

Cytosol trafficking is a limiting step of viral infection or DNA delivery. Starting from the cell surface, most viruses have to travel through a crowded and risky environment in order to reach a small nuclear pore. This work is dedicated to estimating the probability p N of a viral arrival success and, in that case, the mean time τ N it takes. Viral movement is described by a stochastic equation, containing both a drift and a Brownian component. The drift part represents the movement along microtubules, while the Brownian component corresponds to the free diffusion. The success of a viral infection is limited by a killing activity occurring inside the cytoplasm. We model the killing activity by a steady state killing rate k. Because nuclear pores occupy a small fraction of the nuclear area, we use this property to obtain asymptotic estimates of p N and τ N as a function of the diffusion constant D, the amplitude of the drift B and the killing rate k. [ABSTRACT FROM AUTHOR]

We model the motion of a receptor on the membrane surface of a synapse as free Brownian motion in a planar domain with intermittent trappings in and escapes out of corrals with narrow openings. We compute the mean confinement time of the Brownian particle in the asymptotic limit of a narrow opening and calculate the probability to exit through a given small opening, when the boundary contains more than one. Using this approach, it is possible to describe the Brownian motion of a random particle in an environment containing domains with small openings by a coarse grained diffusion process. We use the results to estimate the confinement time as a function of the parameters and also the time it takes for a diffusing receptor to be anchored at its final destination on the postsynaptic membrane, after it is inserted in the membrane. This approach provides a framework for the theoretical study of receptor trafficking on membranes. This process underlies synaptic plasticity, which relates to learning and memory. In particular, it is believed that the memory state in the brain is stored primarily in the pattern of synaptic weight values, which are controlled by neuronal activity. At a molecular level, the synaptic weight is determined by the number and properties of protein channels (receptors) on the synapse. The synaptic receptors are trafficked in and out of synapses by a diffusion process. Following their synthesis in the endoplasmic reticulum, receptors are trafficked to their postsynaptic sites on dendrites and axons. In this model the receptors are first inserted into the extrasynaptic plasma membrane and then random walk in and out of corrals through narrow openings on their way to their final destination. [ABSTRACT FROM AUTHOR]

The distribution of voltage in sub-micron cellular domains remains poorly understood. In neurons, the voltage results from the difference in ionic concentrations which are continuously maintained by pumps and exchangers. However, it not clear how electro-neutrality could be maintained by an excess of fast moving positive ions that should be counter balanced by slow diffusing negatively charged proteins. Using the theory of electro-diffusion, we study here the voltage distribution in a generic domain, which consists of two concentric disks (resp. ball) in two (resp. three) dimensions, where a negative charge is fixed in the inner domain. When global but not local electro-neutrality is maintained, we solve the Poisson–Nernst–Planck equation both analytically and numerically in dimension 1 (flat) and 2 (cylindrical) and found that the voltage changes considerably on a spatial scale which is much larger than the Debye screening length, which assumes electro-neutrality. The present result suggests that long-range voltage drop changes are expected in neuronal microcompartments, probably relevant to explain the activation of far away voltage-gated channels located on the surface membrane. [ABSTRACT FROM AUTHOR]

The time for a DNA sequence to find its homologous counterpart depends on a long random search inside the cell nucleus. Using polymer models, we compute here the mean first encounter time (MFET) between two sites located on two different polymer chains and confined locally by potential wells. We find that reducing tethering forces acting on the polymers results in local decondensation, and numerical simulations of the polymer model show that these changes are associated with a reduction of the MFET by several orders of magnitude. We derive here new asymptotic formula for the MFET, confirmed by Brownian simulations. We conclude from the present modeling approach that the fast search for homology is mediated by a local chromatin decondensation due to the release of multiple chromatin tethering forces. The present scenario could explain how the homologous recombination pathway for double-stranded DNA repair is controlled by its random search step. [ABSTRACT FROM AUTHOR]

Polymer models are used to describe chromatin, which can be folded at different spatial scales by binding molecules. By folding, chromatin generates loops of various sizes. We present here a statistical analysis of the randomly cross-linked (RCL) polymer model, where monomer pairs are connected randomly, generating a heterogeneous ensemble of chromatin conformations. We obtain asymptotic formulas for the steady-state variance, encounter probability, the radius of gyration, instantaneous displacement, and the mean first encounter time between any two monomers. The analytical results are confirmed by Brownian simulations. Finally, the present results are used to extract the mean number of cross links in a chromatin region from conformation capture data. [ABSTRACT FROM AUTHOR]

The first of N identical independently distributed (i.i.d.) Brownian trajectories that arrives to a small target sets the time scale of activation, which in general is much faster than the arrival to the target of a single trajectory only. Analytical asymptotic expressions for the minimal time is notoriously difficult to compute in general geometries. We derive here asymptotic laws for the probability density function of the first and second arrival times of a large number N of i.i.d. Brownian trajectories to a small target in 1, 2 and 3-dimensions and study their range of validity by stochastic simulations. The results are applied to activation of biochemical pathways in cellular biology. [ABSTRACT FROM AUTHOR]

Can the cell’s perception of time be expressed through the length of the shortest telomere? To address this question, we analyze an asymmetric random walk that models telomere length for each division that can decrease by a fixed length a or, if recognized by a polymerase, it increases by a fixed length b  ≫  a . Our analysis of the model reveals two phases, first, a determinist drift of the length toward a quasi-equilibrium state, and second, persistence of the length near an attracting state for the majority of divisions. The measure of stability of the latter phase is the expected number of divisions at the attractor (”lifetime”) prior to crossing a threshold T that model senescence. Using numerical simulations, we further study the distribution of times for the shortest telomere to reach the threshold T . We conclude that the telomerase regulates telomere stability by creating an effective potential barrier that separates statistically the arrival time of the shortest from the next shortest to T . The present model explains how random telomere dynamics underlies the extension of cell survival time. [ABSTRACT FROM AUTHOR]

We report here new electrical laws, derived from nonlinear electrodiffusion theory, about the effect of the local geometrical structure, such as curvature, on the electrical properties of a cell. We adopt the Poisson-Nernst-Planck equations for charge concentration and electric potential as a model of electrodiffusion. In the case at hand, the entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson's equation. We construct an asymptotic approximation for certain singular limits to the steady-state solution in a ball with an attached cusp-shaped funnel on its surface. As the number of charge increases, they concentrate at the end of cusp-shaped funnel. These results can be used in the design of nanopipettes and help to understand the local voltage changes inside dendrites and axons with heterogeneous local geometry. [ABSTRACT FROM AUTHOR]

The electro-diffusion of ions is often described by the Poisson–Nernst–Planck (PNP) equations, which couple nonlinearly the charge concentration and the electric potential. This model is used, among others, to describe the motion of ions in neuronal micro-compartments. It remains at this time an open question how to determine the relaxation and the steady state distribution of voltage when an initial charge of ions is injected into a domain bounded by an impermeable dielectric membrane. The purpose of this paper is to construct an asymptotic approximation to the solution of the stationary PNP equations in a d -dimensional ball ( d = 1 , 2 , 3 ) in the limit of large total charge. In this geometry the PNP system reduces to the Liouville–Gelfand–Bratú (LGB) equation, with the difference that the boundary condition is Neumann, not Dirichlet, and there is a minus sign in the exponent of the exponential term. The entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson’s equation. These differences replace attraction by repulsion in the LGB equation, thus completely changing the solution. We find that the voltage is maximal in the center and decreases toward the boundary. We also find that the potential drop between the center and the surface increases logarithmically in the total number of charges and not linearly, as in classical capacitance theory. This logarithmic singularity is obtained for d = 3 from an asymptotic argument and cannot be derived from the analysis of the phase portrait. These results are used to derive the relation between the outward current and the voltage in a dendritic spine, which is idealized as a dielectric sphere connected smoothly to the nerve axon by a narrow neck. This is a fundamental microdomain involved in neuronal communication. We compute the escape rate of an ion from the steady density in a ball, which models a neuronal spine head, to a small absorbing window in the sphere. We predict that the current is defined by the narrow neck that is connected to the sphere by a small absorbing window, as suggested by the narrow escape theory, while voltage is controlled by the PNP equations independently of the neck. [ABSTRACT FROM AUTHOR]

We consider the escape of a planar diffusion process from the domain of attraction O of a stable focus of the drift in the limit of small diffusion. The boundary O of O is an unstable limit cycle of the drift, and the focus is close to the limit cycle. A new phenomenon of oscillatory decay of the peaks of the survival probability of the process in O emerges for a specific distance of the focus to the boundary which depends on the amplitude of the diffusion coefficient. We show that the phenomenon is due to the complex-valued second eigenvalue 2(O) of the Fokker-Planck operator with Dirichlet boundary conditions. The dominant oscillation frequency, 1/Im{2(O)}, is independent of the relative noise strength. The density of exit points on O is concentrated in a small arc of O closest to the focus. When the focus approaches the boundary, the principal eigenvalue, the reciprocal of which is the mean escape time, does not necessarily decay exponentially as the amplitude of the noise strength decays. The oscillatory escape is manifested in a mathematical model of neural networks with synaptic depression. Oscillation peaks are identified 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 experiments and simulations of neural networks. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the mean first time that two monomers, located on the same polymer, encounter in a confined microdomain. Approximating the confined geometry by a harmonic potential well, we obtain an asymptotic expression for the mean first encounter time (MFETC) as a function of the radius ε around one monomer. By studying the end-to-end distance of the polymer in a ball using the Edwards’ formalism, we derive an other estimation of the MFETC. We validate the asymptotic formulas using Brownian simulations and derive their range of validity in terms of the polymer length. We apply the present models to compute the mean time for a gene located far away from a promoter site to be activated during looping in confined genomic territories. [ABSTRACT FROM AUTHOR]

In section 5.1 entitled "The shortest NEP from a bounded domain in prviously. [ABSTRACT FROM AUTHOR]

Using a novel theoretical approach, we study the mean first-encounter time (MFET) between the two ends of a polymer. Previous approaches used various simplifications that reduced the complexity of the problem, leading, however, to incompatible results. We construct here for the first time a general theory that allows us to compute the MFET. The method is based on estimating the mean time for a Brownian particle to reach a narrow domain in the polymer configuration space. In dimension two and three, we find that the MFET depends mainly on the first eigenvalue of the associated Fokker-Planck operator and provide precise estimates that are confirmed by Brownian simulations. Interestingly, although many time scales are involved in the encounter process, its distribution can be well approximated by a single exponential, which has several consequences for modeling chromosome dynamics in the nucleus. Another application of our result is computing the mean time for a DNA molecule to form a closed loop (when its two ends meet for the first time). [ABSTRACT FROM AUTHOR]

We present a new approach to investigating the dynamics of loop formation in a very crude polymer model and estimate the mean first time for the two ends to meet. This time depends on the number of monomers N. We obtain analytical formulas when N=3, 4 in dimension two and an asymptotic formula when N is large. Our analysis is confirmed by stochastic simulations. [ABSTRACT FROM AUTHOR]

Delivery of nucleic acid to the cell nucleus is a fundamental step in gene therapy. In this review of modeling drug and gene delivery, we focus on the particular stage of plasmid DNA or virus cytoplasmic trafficking. A challenging problem is to quantify the success of this limiting stage. We present some models and simulations of plasmid trafficking and of the limiting phase of DNA-polycation escape from an endosome and discuss virus cytoplasmic trafficking. The models can be used to assess the success of viral escape from endosomes, to quantify the early step of viral-cell infection, and to propose new simulation tools for designing new hybrid-viruses as synthetic vectors. [ABSTRACT FROM AUTHOR]

We estimate the mean first time, called the mean rotation time (MRT), for a planar random polymer to wind around a point. This polymer is modeled as a collection of n rods, each of them being parameterized by a Brownian angle. We are led to study the sum of i.i.d. imaginary exponentials with one dimensional Brownian motions as arguments. We find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function. Finally, we obtain an asymptotic formula for the MRT, whose leading order term depends on $\sqrt{n}$ and, interestingly, depends weakly on the mean initial configuration. Our analytical results are confirmed by Brownian simulations. [ABSTRACT FROM AUTHOR]

Modeling in cellular biology benefits greatly from quantitative analysis that arise from the theory of diffusion and chemical reactions. Recent progress in single particle imaging enables the visualization of viral trajectories evolving in the cytoplasm. Biophysical models and mathematical analysis have been developed to unravel the complexity of single viral trajectories. We review here models of active motion of viruses along the cytoskeleton as well as their diffusion. We present resent efforts to estimate global trafficking properties, such as the probability and the mean time for a viral particle to reach a small nuclear pore. However, most signaling pathways involved in controlling viral motion remain undescribed and should be the goal of future modeling efforts. [Copyright &y& Elsevier]

The study of the diffusive motion of ions or molecules in confined biological microdomains requires the derivation of the explicit dependence of quantities, such as the decay rate of the population or the forward chemical reaction rate constant on the geometry of the domain. Here, we obtain this explicit dependence for a model of a Brownian particle (ion, molecule, or protein) confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. We call the calculation of the mean escape time the narrow escape problem. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem. Here, we present asymptotic formulas for the mean escape time in several cases, including regular domains in two and three dimensions and in some singular domains in two dimensions. The mean escape time comes up in many applications, because it represents the mean time it takes for a molecule to hit a target binding site. We present several applications in cellular biology: calcium decay in dendritic spines, a Markov model of multicomponent chemical reactions in microdomains, dynamics of receptor diffusion on the surface of neurons, and vesicle trafficking inside a cell. [ABSTRACT FROM AUTHOR]

A Brownian particle with diffusion coefficient D is confined to a bounded domain Ω by a reflecting boundary, except for a small absorbing window $$\partial\Omega_a$$ . The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. In the three-dimensional case, we construct an asymptotic approximation when the window is an ellipse, assuming the large semi axis a is much smaller than $$|\Omega|^{1/3}$$ ( $$|\Omega|$$ is the volume), and show that the mean escape time is $$E\tau\sim{\frac{|\Omega|}{2\pi Da}} K(e)$$ , where e is the eccentricity 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 $$E\tau\sim{\frac{|\Omega|}{4aD}}$$ , which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion $$E\tau={\frac{|\Omega|}{4aD}} [1+\frac{a}{R} \log \frac{R}{a} + O(\frac{a}{R})]$$ . This result is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function. If Ω is a two-dimensional bounded Riemannian manifold with metric g and $$\varepsilon=|\partial\Omega_a|_g/|\Omega|_g\ll1$$ , we show that $$E\tau ={\frac{|\Omega|_g}{D\pi}}[\log{\frac{1}{\varepsilon}}+O(1)]$$ . This result is applicable to diffusion in membrane surfaces. [ABSTRACT FROM AUTHOR]

We consider the narrow escape problem in two-dimensional Riemannian manifolds (with a metric g) with corners and cusps, in an annulus, and on a sphere. Specifically, we calculate the mean time it takes a Brownian particle diffusing in a domain Ω to reach an absorbing window when the ratio $${\varepsilon= {\frac{|\partial \Omega_a|_g}{|\partial \Omega|_g}}}$$ between the absorbing window and the otherwise reflecting boundary is small. If the boundary is smooth, as in the cases of the annulus and the sphere, the leading term in the expansion is the same as that given in part I of the present series of papers, however, when it is not smooth, the leading order term is different. If the absorbing window is located at a corner of angle α, then $$E\tau = { \frac{|\Omega|_g}{\alpha D}}[\log{\frac{1}{\varepsilon}}+O(1)],$$ 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 $$E\tau ={\frac{|\Omega|}{(d^{-1}-1)D}}(\frac{1}{\varepsilon} +O(1))$$ , where $$d<1$$ is the ratio of the radii. For the smooth boundary case, we calculate the next term of the expansion for the annulus and the sphere. 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 $$\log{\frac{1}{\varepsilon}}$$ is not necessarily large, even when $$\varepsilon = {\frac{|\partial \Omega_a|_g}{|\partial \Omega|_g}}$$ is small. In these two problems there are additional parameters that can be small, such as the ratio δ of the radii of the annulus. The contributions of these parameters to the expansion of the mean escape time are also logarithmic. In the case of the annulus the mean escape time is $$E\tau = {\frac{|\Omega|_g}{\pi D} [\log\frac{1}{\varepsilon}+\frac 12\log\frac{1}{\delta}+O(1)]}$$ . [ABSTRACT FROM AUTHOR]

We consider Brownian motion in a circular disk Ω, whose boundary $$\partial\Omega$$ is reflecting, except for a small arc, $$\partial\Omega_a$$ , which is absorbing. As $$\varepsilon=|\partial\Omega_a|/|\partial \Omega|$$ decreases to zero the mean time to absorption in $$\partial\Omega_a$$ , denoted $$E\tau$$ , becomes infinite. The narrow escape problem is to find an asymptotic expansion of $$E\tau$$ for $$\varepsilon\ll1$$ . 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 domain, $$E\tau = {\frac{|\Omega|}{D\pi}}\big[\log{\frac{1}{\varepsilon}}+O(1)\big],$$ ( $$D$$ 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 $$E[\tau\,|\,{\boldmath x}(0)={\bf 0}]={\frac{R^2}{D}}\big[\log{\frac{1}{\varepsilon}}+ \log 2 +{ \frac{1}{4}} + O(\varepsilon)\big]$$ . The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because $$\log{\frac{1}{\varepsilon}}$$ is not necessarily large, even when ε is small. We also find the singular behavior of the probability flux profile into $$\partial\Omega_a$$ at the endpoints of $$\partial\Omega_a$$ , and find the value of the flux near the center of the window. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The cellular membrane is very heterogenous and enriched with high-density regions forming microdomains, as revealed by single particle tracking experiments. However the organization of these regions remain unexplained. We determine here the biophysical properties of these regions, when described as a basin of attraction. We develop two methods to recover the dynamics and local potential wells (field of force and boundary). The first method is based on the local density of points distribution of trajectories, which differs inside and outside the wells. The second method focuses on recovering the drift field that is convergent inside wells and uses the transient field to determine the boundary. Finally, we apply these two methods to the distribution of trajectories recorded from voltage gated calcium channels and phospholipid anchored GFP in the cell membrane of hippocampal neurons and obtain the size and energy of high-density regions with a nanometer precision. [ABSTRACT FROM AUTHOR]