Calculation Of First Passage Time Research Articles

Noise-induced bistability in a simple mutual inhibition system.

In this study, we study noise-induced bistability in a simple bivariate mutual inhibition system with slow fluctuating responses to external signals. We give a general condition that the marginal stationary probability density of one of the two variables experiences a transition from a unimodal shape to a bimodal one. We show that the transition occurs even when the stationary probability density of the response to external signals is monotone. The mechanism for the transition is investigated in terms of the calculation of the mean first passage time. We also discuss the genericity of the transition mechanism.

Optimizing microtubule arrangements for rapid cargo capture

Cellular functions such as autophagy, cell signaling, and vesicular trafficking involve the retrograde transport of motor-driven cargo along microtubules. Typically, newly formed cargo engages in slow undirected movement from its point of origin before attaching to a microtubule. In some cell types, cargo destined for delivery to the perinuclear region relies on capture at dynein-enriched loading zones located near microtubule plus ends. Such systems include extended cell regions of neurites and fungal hyphae, where the efficiency of the initial diffusive loading process depends on the axial distribution of microtubule plus ends relative to the initial cargo position. We use analytic mean first-passage time calculations and numerical simulations to model diffusive capture processes in tubular cells, exploring how the spatial arrangement of microtubule plus ends affects the efficiency of retrograde cargo transport. Our model delineates the key features of optimal microtubule arrangements that minimize mean cargo capture times. Namely, we show that configurations with a single microtubule plus end abutting the distal tip and broadly distributed other plus ends allow for efficient capture in a variety of different scenarios for retrograde transport. Live-cell imaging of microtubule plus ends in Aspergillus nidulans hyphae indicates that their distributions exhibit these optimal qualitative features. Our results highlight important coupling effects between the distribution of microtubule tips and retrograde cargo transport, providing guiding principles for the spatial arrangement of microtubules within tubular cell regions.

First-passage time theory of activated rate chemical processes in electronic molecular junctions.

Confined nanoscale spaces, electric fields, and tunneling currents make the molecular electronic junction an experimental device for the discovery of new out-of-equilibrium chemical reactions. Reaction-rate theory for current-activated chemical reactions is developed by combining the Keldysh nonequilibrium Green's function treatment of electrons, Fokker-Planck description of the reaction coordinate, and Kramers first-passage time calculations. The nonequilibrium Green's functions (NEGF) provide an adiabatic potential as well as a diffusion coefficient and temperature with local dependence on the reaction coordinate. Van Kampen's Fokker-Planck equation, which describes a Brownian particle moving in an external potential in an inhomogeneous medium with a position-dependent friction and diffusion coefficient, is used to obtain an analytic expression for the first-passage time. The theory is applied to several transport scenarios: a molecular junction with a single reaction coordinate dependent molecular orbital and a model diatomic molecular junction. We demonstrate the natural emergence of Landauer's blowtorch effect as a result of the interplay between the configuration dependent viscosity and diffusion coefficients. The resultant localized heating in conjunction with the bond-deformation due to current-induced forces is shown to be the determining factors when considering chemical reaction rates, each of which results from highly tunable parameters within the system.

First passage time in multistep stochastic processes with applications to dust charging.

An approach was developed to describe the first passage time (FPT) in multistep stochastic processes with discrete states governed by a master equation (ME). The approach is an extension of the totally absorbing boundary approach given for calculation of FPT in one-step processes [N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier Science Publishers, North Holland, Amsterdam, 2007)] to include multistep processes where jumps are not restricted to adjacent sites. In addition, a Fokker-Planck equation (FPE) was derived from the multistep ME, assuming the continuity of the state variable. The developed approach and an FPE based approach [C. W. Gardiner, Handbook of Stochastic Methods, 3rd ed. (Springer-Verlag, New York, 2004)] were used to find the mean first passage time (MFPT) of the transition between the negative and positive stable macrostates of dust grain charge when the charging process was bistable. The dust was in a plasma and charged by collecting ions and electrons, and emitting secondary electrons. The MFPTs for the transitioning of grain charge from one macrostate to the other were calculated by the two approaches for a range of grain sizes. Both approaches produced very similar results for the same grain except for when it was very small. The difference between MFPTs of two approaches for very small grains was attributed to the failure of the charge continuity assumption in the FPE description. For a given grain, the MFPT for a transition from the negative macrostate to the positive one was substantially larger than that for a transition in a reverse order. The normalized MFPT for a transition from the positive to the negative macrostate showed little sensitivity to the grain radius. For a reverse transition, with the increase of the grain radius, it dropped first and then increased. The probability density function of FPT was substantially wider for a transition from the positive to the negative macrostate, as compared to a reverse transition.

Bubbles in water under stretch-induced cavitation.

When a finite sample of water experiences tension, it may develop voids (bubbles). We present here a result for the work (Helmholtz free energy change) that needs to be done for the creation of a bubble in fixed volume of water under tension and show that this result depends on the general form of stress-strain relationship. We observe that it is very important to include the curvature-dependent surface tension into consideration in order to explain bubble stability. The analytical result we obtained for the free energy allows us to make prediction for the values of critical and stable radii of the bubbles. We also performed simulations on the TIP4P/2005 water model and observed creation of bubbles in water under stretch. Combining analytical results obtained from our thermodynamic description with the results from computer simulations allowed us to determine the two parameters that describe the curvature-dependent surface tension and also to find the values of critical and stable bubble radii. We also determined the values of critical bubble radii by using mean first-passage time calculations.

A Mathematical Theory of Optimal Milestoning (with a Detour via Exact Milestoning)

AbstractMilestoning is a computational procedure that reduces the dynamics of complex systems to memoryless jumps between intermediates, or milestones, and only retains some information about the probability of these jumps and the time lags between them. Here we analyze a variant of this procedure, termed optimal milestoning, which relies on a specific choice of milestones to capture exactly some kinetic features of the original dynamical system. In particular, we prove that optimal milestoning permits the exact calculation of the mean first passage times (MFPT) between any two milestones. In so doing, we also analyze another variant of the method, called exact milestoning, which also permits the exact calculation of certain MFPTs, but at the price of retaining more information about the original system's dynamics. Finally, we discuss importance sampling strategies based on optimal and exact milestoning that can be used to bypass the simulation of the original system when estimating the statistical quantities used in these methods.© 2017 Wiley Periodicals, Inc.

Spatial pattern formation in microtubule post-translational modifications and the tight localization of motor-driven cargo

Microtubule (MT) "age" can be interpreted as nucleotide state, lattice defects, or post-translational modification (PTM) such as acetylation and detyrosination. In all three cases, these have been recently shown to have functionally-important effects on the dynamics of MT arrays, and can present spatial and temporal heterogeneity. While mathematical models for MT array densities are well-established, here we present equations describing MT age, defined as the mean time since the MT's building blocks (tubulin) were polymerized from their soluble dimer state. We derive the age equations using a mean first-passage time calculation and two complementary approaches: The continuum limit of spatial discretization model, and an adjoint operator approach. These equations can recapitulate the observation that the oldest (most de-tyrosinated) tubulin in axons is near the middle of axons during neuronal development in chick embryos. Furthermore, PTMs influence motor kinetics up to approximately twofold for off-rates and velocities. Our simulations demonstrate that this relatively weak dependence of motor kinetics is sufficient to target motor cargo to a specific location along the array. This localization is tightly peaked in a way that magnifies the relatively small signal of PTM spatial heterogeneity. Thus, MT age can produce long-range spatial patterning without feedbacks or diffusing signals.

Probability distribution function for reorientations in Maier–Saupe potential

Exact analytic solution for the probability distribution function of the non-inertial rotational diffusion equation, i.e., of the Smoluchowski one, in a symmetric Maier–Saupe uniaxial potential of mean torque is obtained via the confluent Heun’s function. Both the ordinary Maier–Saupe potential and the double-well one with variable barrier width are considered. Thus, the present article substantially extends the scope of the potentials amenable to the treatment by reducing Smoluchowski equation to the confluent Heun’s one. The solution is uniformly valid for any barrier height. We use it for the calculation of the mean first passage time. Also the higher eigenvalues for the relaxation decay modes in the case of ordinary Maier–Saupe potential are calculated. The results obtained are in full agreement with those of the approach developed by Coffey, Kalmykov, Déjardin and their coauthors in the whole range of barrier heights.

Asymptotic analysis of narrow escape problems in nonspherical three-dimensional domains.

Narrow escape problems consider the calculation of the mean first passage time (MFPT) for a particle undergoing Brownian motion in a domain with a boundary that is everywhere reflecting except for at finitely many small holes. Asymptotic methods for solving these problems involve finding approximations for the MFPT and average MFPT that increase in accuracy with decreasing hole sizes. While relatively much is known for the two-dimensional case, the results available for general three-dimensional domains are rather limited. This paper addresses the problem of finding the average MFPT for a class of three-dimensional domains bounded by the level surface of an orthogonal coordinate system. In particular, this class includes spheroids and other solids of revolution. The primary result presented is a two-term asymptotic expansion for the average MFPT of such domains containing an arbitrary number of holes. Steps are taken towards finding higher-order asymptotic expansions for both the average MFPT and the MFPT in these domains. The results for the average MFPT are compared to full numerical calculations performed with the comsol Multiphysics finite element solver for three distinct domains: prolate and oblate spheroids and biconcave disks. This comparison shows good agreement with the proposed two-term expansion of the average MFPT in the three domains.

Conditional Mean First Passage Times to Small Traps in a 3-D Domain with a Sticky Boundary: Applications to T Cell Searching Behavior in Lymph Nodes

Calculating the time required for a diffusing object to reach a small target within a larger domain is a feature of a large class of modeling and simulation efforts in biology. Here, we are motivated by the motion of a T cell of the immune system seeking a particular antigen-presenting cell within a large lymph node. The precise nature of the cell motion at the outer boundary of the lymph node is not completely understood in terms of how cells choose to remain within a given lymph node, or exit. In previous work, we and others have studied diffusive motion to a small trap. We extend this previous work to analyze models where the diffusing object may exit the outer boundary of the domain (in this case, the lymph node). This is modeled by a Robin boundary condition on the surface of the lymph node. For the general problem of small traps inside a three-dimensional domain that has a partially sticky or absorbent domain boundary, the method of matched asymptotic expansions is used to calculate the mean and variance of the conditional first passage time for the T cell to reach a specific target trap. Our results are illustrated explicitly for the idealized situation of a spherical lymph node containing small spherically shaped traps, and are verified for a radially symmetric geometry with one trap at the origin where exact solutions are available. Mathematically, our analysis extends previous work on the calculation of the mean first passage time by allowing for a sticky boundary and by calculating conditional statistics of the diffusion process. Finally, our results are interpreted and applied to the context of T cell biology.

1D Tight-Binding Models Render Quantum First Passage Time “Speakable”

The calculation of First Passage Time (moreover, even its probability density in time) has so far been generally viewed as an ill-posed problem in the domain of quantum mechanics. The reasons can be summarily seen in the fact that the quantum probabilities in general do not satisfy the Kolmogorov sum rule: the probabilities for entering and non-entering of Feynman paths into a given region of space-time do not in general add up to unity, much owing to the interference of alternative paths. In the present work, it is pointed out that a special case exists (within quantum framework), in which, by design, there exists one and only one available path (i.e., door-way) to mediate the (first) passage -no alternative path to interfere with. Further, it is identified that a popular family of quantum systems - namely the 1d tight binding Hamiltonian systems - falls under this special category. For these model quantum systems, the first passage time distributions are obtained analytically by suitably applying a method originally devised for classical (stochastic) mechanics (by Schroedinger in 1915). This result is interesting especially given the fact that the tight binding models are extensively used in describing everyday phenomena in condense matter physics.

Exact solution of Smoluchowski’s equation for reorientational motion in Maier–Saupe potential

The analytic treatment of the non-inertial rotational diffusion equation, i.e., of Smoluchowski’s one (SE), in a symmetric genuinely double-well Maier–Saupe uniaxial potential of mean torque is considered. Such potential may find applications to reorientations of the fragments of structure in polymers and proteins. We obtain the exact solution of SE via the confluent Heun’s function. The solution is uniformly valid for any barrier height. We apply the obtained solution to the calculation of the mean first passage time and the longitudinal correlation time and obtain their precise dependence on the barrier height. In the intermediate to high barrier (low temperature) region the results of our approach are in full agreement with those of the approach developed by Coffey, Kalmykov, Déjardin and their coauthors. In the low barrier (high temperature) region our results noticeably distinguish from the predictions of the literature formula and give appreciably greater values for the transition rates from the potential well. The reason is that the above mentioned formula is obtained in the stationary limit. We conclude that for very small barrier heights the transient dynamics plays a crucial role and has to be taken into account explicitly. When this requirement is satisfied (as, e.g, at the calculation of the longitudinal correlation time) we obtain absolute identity of our results with the literature formula in the whole range of barrier heights. The drawbacks of our approach are its applicability only to the symmetric potential and its inability to yield an analytical expression for the smallest non-vanishing eigenvalue.

The Single-Fiber Collision Rate and Filtration Efficiency for Nanoparticles II: Extension to Arbitrary-Shaped Particles

We extend the equations for the dimensionless collision kernel and filtration efficiency, attained previously via mean first-passage time (MFPT) calculations, to particles of arbitrary shape. Specifically, we show that the regression equations for the dimensionless collision rate found considering particle-fiber collisions driven by simultaneous diffusion and interception remain valid for non-spherical particles, provided that an appropriate collision length scale for the non-spherical particle (L) is defined and incorporated into the definitions of the dimensionless collision rate (H) and the diffusive Knudsen number (KnD). Regression equations are provided to calculate this length scale for quasifractal aggregates of varying fractal dimension, as well as cylinders. MFPT calculations reveal that, over ∼5 orders of magnitude in H, these regression equations for the collision length are valid. Furthermore, using the previously attained proportionality between the predicted dimensionless collision rate and ...

The Single-Fiber Collision Rate and Filtration Efficiency for Nanoparticles I: The First-Passage Time Calculation Approach

We describe an approach to filtration-efficiency calculations as an alternative to the traditional depth filtration theory. The new approach involves linking the single-fiber efficiency to the collision rate coefficient/kernel between nanoparticles and fibers, and correspondingly inferring the collision kernel via dimensionless mean first-passage time (MFPT) calculations. This method has the advantage of easily incorporating the influences of particle diffusion, inertia, and particle size; therefore, all filtration mechanisms can be considered simultaneously. Through non-dimensionalization of the equation of motion for a particle in MFPT calculations (the Langevin equation), it is shown that both the single-fiber efficiency Ef and dimensionless particle-fiber collision kernel, H, are functions of the ratio of particle radius to filter-fiber radius, R, the solid volume fraction in the filter, Vf, the ratio of particle persistence distance to the particle-filter collision distance, KnD (the diffusive Knudse...

Path integrals and large deviations in stochastic hybrid systems.

We construct a path-integral representation of solutions to a stochastic hybrid system, consisting of one or more continuous variables evolving according to a piecewise-deterministic dynamics. The differential equations for the continuous variables are coupled to a set of discrete variables that satisfy a continuous-time Markov process, which means that the differential equations are only valid between jumps in the discrete variables. Examples of stochastic hybrid systems arise in biophysical models of stochastic ion channels, motor-driven intracellular transport, gene networks, and stochastic neural networks. We use the path-integral representation to derive a large deviation action principle for a stochastic hybrid system. Minimizing the associated action functional with respect to the set of all trajectories emanating from a metastable state (assuming that such a minimization scheme exists) then determines the most probable paths of escape. Moreover, evaluating the action functional along a most probable path generates the so-called quasipotential used in the calculation of mean first passage times. We illustrate the theory by considering the optimal paths of escape from a metastable state in a bistable neural network.

Water self-diffusion at the surface of silica glasses: effect of hydrophilic to hydrophobic transition

We study water dynamics at a silica aqueous interface. Both hydrophilic (hydroxylated) surfaces and hydrophobic surfaces (dehydroxylated upon irradiation) have been generated from atomistic simulations. A new method for the calculation of the normal self-diffusion coefficients based on the calculation of mean first passage times is proposed. It uses the Smoluchowski theory of Brownian motion and it takes proper account of the layering of the molecules. In the case of parallel self-diffusion coefficients, a decrease is found compared to the bulk values. It can be described in terms of hydrodynamic boundary conditions induced by the surface confinement. This hydrodynamic explanation is not enough to interpret the case of normal self-diffusion coefficients for which an important diminution is found. Normal self-diffusion coefficients appear to depend strongly on the hydrophilicity of the surface. They tend towards their bulk value only at long distances from the surfaces. The first layer of water molecules is found to be partially adsorbed.

When is the next extending of Fick-Jacobs equation necessary?

Applicability of the effective one-dimensional equations, such as Fick-Jacobs equation and its extensions, describing diffusion of particles in 2D or 3D channels with varying cross section A(x) along the longitudinal coordinate x, is studied. The leading nonstationary correction to Zwanzig-Reguera-Rubí equation [R. Zwanzig, J. Phys. Chem. 96, 3926 (1992); D. Reguera and J. M. Rubí, Phys. Rev. E 64, 061106 (2001)] is derived and tested on the exactly solvable model, diffusion in a 2D linear cone. The effects of such correction are demonstrated and discussed on elementary nonstationary processes, a time dependent perturbation of the stationary flow and calculation of the mean first passage time.

Design and Analysis of Saturated Ring Oscillators Based on the Random Mid-Point Voltage Concept

As technology scales, a lower supply voltage means transistors in ring oscillators frequently switch between triode and saturation. A model based on random mid-point voltage is developed in this brief. It reduces the number of first-passage time calculations. For instance, when applied to a current-starved inverter delay cell, the number of calculations is reduced from 5 to 1. Moreover, when applied to a differential pair delay cell, new design insights develop. These insights result in improved phase noise (PN) via a combination of reduced input pair size and the addition of extra poly capacitance at the output. Measured PN on oscillators fabricated in 0.18-μm CMOS, which oscillate at 448 and 473 MHz, shows an improvement of 6 dBc/Hz. In both types of delay cells, the model agrees reasonably well with measurement results.

Expanded Koch networks: structure and trapping time of random walks

Based on the Koch network constructed using Koch fractals, we proposed a class of expanded Koch networks in this paper. The original triangle is replaced by r-polygon, and each node generates m sub r-polygons by every step, which makes the Koch network more general. We studied the structure and properties of the networks. The exact analytical result of the degree distribution, clustering coefficient and average path length were obtained. When parameters m and r satisfy some certain conditions, the networks follow a power-law distribution and have a small average path length. Finally, we introduced the random walk on the network. Our discussions focused on the trapping problem, particularly the calculation and derivation of mean first passage time (MFPT) and global mean first passage time (GMFPT). In addition, we also gave the relationship between the above results and the network size.

Mean first-passage time calculations: comparison of the deterministic Hill’s algorithm with Monte Carlo simulations

Accurate determination of mean first-passage times (MFPTs) between any two states of a complex network still attracts considerable attention. Appropriate methods should take into account the discrepancy in MFPTs when a random walker moves first from a source to a target and then in the opposite direction. In addition, it is desirable to allow fast evaluation of mean first-passage times when transition probabilities are allowed to vary over time. For our calculations we make use of Hill’s algorithm, which enables the exact calculation of MFPTs. As we show in this work, when given a fixed distance to travel, the calculation of a particular MFPT depends on the choice of source and target points and their relative position on a lattice. We also demonstrate, when the network contains a relatively low number of cycles, that this deterministic technique provides exact results much faster in comparison to the more standard, but computationally demanding, stochastic Monte Carlo simulation method, where only approximate results can be obtained that are highly dependent on the number of walkers. Therefore, our specific implementation of Hill’s algorithm should facilitate efficient and accurate computation of MFPTs on a variety of network topologies of practical interest to the broad scientific community.