Stochastic Chemical Reactions in Micro-domains

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 Physic

A Path Intergal Approach to Current

Discontinuous initial wave functions or wave functions with discontintuous derivative and with bounded support arise in a natural way in various situations in physics, in particular in measurement theory. The propagation of such initial wave functions is not well described by the Schr\"odinger current which vanishes on the boundary of the support of the wave function. This propagation gives rise to a uni-directional current at the boundary of the support. We use path integrals to define current and uni-directional current and give a direct derivation of the expression for current from the path integral formulation for both diffusion and quantum mechanics. Furthermore, we give an explicit asymptotic expression for the short time propagation of initial wave function with compact support for both the cases of discontinuous derivative and discontinuous wave function. We show that in the former case the probability propagated across the boundary of the support in time $\Delta t$ is $O(\Delta t^{3/2})$ and the initial uni-directional current is $O(\Delta t^{1/2})$. This recovers the Zeno effect for continuous detection of a particle in a given domain. For the latter case the probability propagated across the boundary of the support in time $\Delta t$ is $O(\Delta t^{1/2})$ and the initial uni-directional current is $O(\Delta t^{-1/2})$. This is an anti-Zeno effect. However, the probability propagated across a point located at a finite distance from the boundary of the support is $O(\Delta t)$. This gives a decay law.Comment: 17 pages, Late

Measurement as Absorption of Feynman Trajectories: Collapse of the Wave Function Can be Avoided

We define a measuring device (detector) of the coordinate of quantum particle as an absorbing wall that cuts off the particle's wave function. The wave function in the presence of such detector vanishes on the detector. The trace the absorbed particles leave on the detector is identifies as the absorption current density on the detector. This density is calculated from the solution of Schr\"odinger's equation with a reflecting boundary at the detector. This current density is not the usual Schr\"odinger current density. We define the probability distribution of the time of arrival to a detector in terms of the absorption current density. We define coordinate measurement by an absorbing wall in terms of 4 postulates. We postulate, among others, that a quantum particle has a trajectory. In the resulting theory the quantum mechanical collapse of the wave function is replaced with the usual collapse of the probability distribution after observation. Two examples are presented, that of the slit experiment and the slit experiment with absorbing boundaries to measure time of arrival. A calculation is given of the two dimensional probability density function of a free particle from the measurement of the absorption current on two planes.Comment: 20 pages, latex, no figure

Distributed state estimation in sensor networks with randomly occurring nonlinearities subject to time delays

This is the post-print version of the Article. The official published version can be accessed from the links below - Copyright @ 2012 ACM.This article is concerned with a new distributed state estimation problem for a class of dynamical systems in sensor networks. The target plant is described by a set of differential equations disturbed by a Brownian motion and randomly occurring nonlinearities (RONs) subject to time delays. The RONs are investigated here to reflect network-induced randomly occurring regulation of the delayed states on the current ones. Through available measurement output transmitted from the sensors, a distributed state estimator is designed to estimate the states of the target system, where each sensor can communicate with the neighboring sensors according to the given topology by means of a directed graph. The state estimation is carried out in a distributed way and is therefore applicable to online application. By resorting to the Lyapunov functional combined with stochastic analysis techniques, several delay-dependent criteria are established that not only ensure the estimation error to be globally asymptotically stable in the mean square, but also guarantee the existence of the desired estimator gains that can then be explicitly expressed when certain matrix inequalities are solved. A numerical example is given to verify the designed distributed state estimators.This work was supported in part by the National Natural Science Foundation of China under Grants 61028008, 60804028 and 61174136, the Qing Lan Project of Jiangsu Province of China, the Project sponsored by SRF for ROCS of SEM of China, the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Royal Society of the UK, and the Alexander von Humboldt Foundation of Germany

Ions in Fluctuating Channels: Transistors Alive

Ion channels are proteins with a hole down the middle embedded in cell membranes. Membranes form insulating structures and the channels through them allow and control the movement of charged particles, spherical ions, mostly Na+, K+, Ca++, and Cl-. Membranes contain hundreds or thousands of types of channels, fluctuating between open conducting, and closed insulating states. Channels control an enormous range of biological function by opening and closing in response to specific stimuli using mechanisms that are not yet understood in physical language. Open channels conduct current of charged particles following laws of Brownian movement of charged spheres rather like the laws of electrodiffusion of quasi-particles in semiconductors. Open channels select between similar ions using a combination of electrostatic and 'crowded charge' (Lennard-Jones) forces. The specific location of atoms and the exact atomic structure of the channel protein seems much less important than certain properties of the structure, namely the volume accessible to ions and the effective density of fixed and polarization charge. There is no sign of other chemical effects like delocalization of electron orbitals between ions and the channel protein. Channels play a role in biology as important as transistors in computers, and they use rather similar physics to perform part of that role. Understanding their fluctuations awaits physical insight into the source of the variance and mathematical analysis of the coupling of the fluctuations to the other components and forces of the system.Comment: Revised version of earlier submission, as invited, refereed, and published by journa

Robust synchronization of a class of coupled delayed networks with multiple stochastic disturbances: The continuous-time case

In this paper, the robust synchronization problem is investigated for a new class of continuous-time complex networks that involve parameter uncertainties, time-varying delays, constant and delayed couplings, as well as multiple stochastic disturbances. The norm-bounded uncertainties exist in all the network parameters after decoupling, and the stochastic disturbances are assumed to be Brownian motions that act on the constant coupling term, the delayed coupling term as well as the overall network dynamics. Such multiple stochastic disturbances could reflect more realistic dynamical behaviors of the coupled complex network presented within a noisy environment. By using a combination of the Lyapunov functional method, the robust analysis tool, the stochastic analysis techniques and the properties of Kronecker product, we derive several delay-dependent sufficient conditions that ensure the coupled complex network to be globally robustly synchronized in the mean square for all admissible parameter uncertainties. The criteria obtained in this paper are in the form of linear matrix inequalities (LMIs) whose solution can be easily calculated by using the standard numerical software. The main results are shown to be general enough to cover many existing ones reported in the literature. Simulation examples are presented to demonstrate the feasibility and applicability of the proposed results

Temporal dynamics of tunneling. Hydrodynamic approach

We use the hydrodynamic representation of the Gross -Pitaevskii/Nonlinear Schroedinger equation in order to analyze the dynamics of macroscopic tunneling process. We observe a tendency to a wave breaking and shock formation during the early stages of the tunneling process. A blip in the density distribution appears in the outskirts of the barrier and under proper conditions it may transform into a bright soliton. Our approach, based on the theory of shock formation in solutions of Burgers equation, allows us to find the parameters of the ejected blip (or soliton if formed) including the velocity of its propagation. The blip in the density is formed regardless of the value and sign of the nonlinearity parameter. However a soliton may be formed only if this parameter is negative (attraction) and large enough. A criterion is proposed. An ejection of a soliton is also observed numerically. We demonstrate, theoretically and numerically, controlled formation of soliton through tunneling. The mass of the ejected soliton is controlled by the initial state.Comment: 11 pages, 6 figures, expanded and more detailed verions of the previous submissio

Thermophoresis of Brownian particles driven by coloured noise

The Brownian motion of microscopic particles is driven by the collisions with the molecules of the surrounding fluid. The noise associated with these collisions is not white, but coloured due, e.g., to the presence of hydrodynamic memory. The noise characteristic time scale is typically of the same order as the time over which the particle's kinetic energy is lost due to friction (inertial time scale). We demonstrate theoretically that, in the presence of a temperature gradient, the interplay between these two characteristic time scales can have measurable consequences on the particle long-time behaviour. Using homogenization theory, we analyse the infinitesimal generator of the stochastic differential equation describing the system in the limit where the two characteristic times are taken to zero; from this generator, we derive the thermophoretic transport coefficient, which, we find, can vary in both magnitude and sign, as observed in experiments. Furthermore, studying the long-term stationary particle distribution, we show that particles can accumulate towards the colder (positive thermophoresis) or the warmer (negative thermophoresis) regions depending on the dependence of their physical parameters and, in particular, their mobility on the temperature.Comment: 9 pages, 4 figure

Realistic boundary conditions for stochastic simulations of reaction-diffusion processes

Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). Such reaction-diffusion processes can be mathematically modelled using either deterministic partial-differential equations or stochastic simulation algorithms. The latter provide a more detailed and precise picture, and several stochastic simulation algorithms have been proposed in recent years. Such models typically give the same description of the reaction-diffusion processes far from the boundary of the simulated domain, but the behaviour close to a reactive boundary (e.g. a membrane with receptors) is unfortunately model-dependent. In this paper, we study four different approaches to stochastic modelling of reaction-diffusion problems and show the correct choice of the boundary condition for each model. The reactive boundary is treated as partially reflective, which means that some molecules hitting the boundary are adsorbed (e.g. bound to the receptor) and some molecules are reflected. The probability that the molecule is adsorbed rather than reflected depends on the reactivity of the boundary (e.g. on the rate constant of the adsorbing chemical reaction and on the number of available receptors), and on the stochastic model used. This dependence is derived for each model.Comment: 24 pages, submitted to Physical Biolog