Modeling ion channel dynamics through reflected stochastic differential equations

Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the "gold standard," but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks. © 2012 American Physical Society.

Pulse dynamics in a three-component system : stability and bifurcations

In this article, we analyze the stability and the associated bifurcations of several types of pulse solutions in a singularly perturbed three-component reaction-diffusion equation that has its origin as a model for gas discharge dynamics. Due to the richness and complexity of the dynamics generated by this model, it has in recent years become a paradigm model for the study of pulse interactions. A mathematical analysis of pulse interactions is based on detailed information on the existence and stability of isolated pulse solutions. The existence of these isolated pulse solutions is established in previous work. Here, the pulse solutions are studied by an Evans function associated to the linearized stability problem. Evans functions for stability problems in singularly perturbed reaction-diffusion models can be decomposed into a fast and a slow component, and their zeroes can be determined explicitly by the NLEP method. In the context of the present model, we have extended the NLEP method so that it can be applied to multi-pulse and multi-front solutions of singularly perturbed reaction-diffusion equations with more than one slow component. The brunt of this article is devoted to the analysis of the stability characteristics and the bifurcations of the pulse solutions. Our methods enable us to obtain explicit, analytical information on the various types of bifurcations, such as saddle-node bifurcations, Hopf bifurcations in which breathing pulse solutions are created, and bifurcations into travelling pulse solutions, which can be both subcritical and supercritical.

Base-induced decomposition of alkyl hydroperoxides in the gas phase. Part 3. Kinetics and dynamics in HO-+CH3OOH, C2H5OOH, and tert-C4H9OOH reactions

The E-CO(2) elimination reactions of alkyl hydroperoxides proceed via abstraction of an (x-hydrogen by a base: X- + (RRHCOOH)-R-1-H-2 -> HX + (RRC)-R-1-C-2=O + HO-. Efficiencies and product distributions for the reactions of the hydroxide anion with methyl, ethyl, and tert-butyl hydroperoxides are studied in the gas phase. On the basis of experiments using three isotopic analogues, HO- + CH3OOH, HO- + CD3OOH, and H18O- + CH3OOH. the overall intrinsic reaction efficiency is determined to be 80% or greater. The E(CO)2 decomposition is facile for these methylperoxide reactions, and predominates over competing proton transfer at the hydroperoxide moiety. The CH3CH2OOH reaction displays a similar E(CO)2 reactivity, whereas proton transfer and the formation of HOO- are the exclusive pathways observed for (CH3)(3)COOH, which has no (x-hydrogen. All results are consistent with the E-CO(2) mechanism, transition state structure, and reaction energy diagrams calculated using the hybrid density functional B3LYP approach. Isotope labeling for HO- + CH3OOH also reveals some interaction between H2O and HO- within the E(CO)2 product complex [H2O center dot center dot center dot CH2=O center dot center dot center dot HO-]. There is little evidence, however. for the formation of the most exothermic products H2O + CH2(OH)O-, which would arise from nuclephilic condensation of CH2=O and HO-. The results suggest that the product dynamics are not totally statistical but are rather direct after the E-CO(2) transition state. The larger HO- + CH3CH2OOH system displays more statistical behavior during complex dissociation.

Simulation of gas-phase nanoparticle dynamics in the plasma-enhanced chemical vapor deposition of carbon nanostructures

The results of 1D simulation of nanoparticle dynamics in the areas adjacent to nanostructured carbon-based films exposed to chemically active complex plasma of CH4 + H2 + Ar gas mixtures are presented. The nanoparticle-loaded near-substrate (including sheath and presheath) areas of a low-frequency (0.5 MHz) inductively coupled plasma facility for the PECVD growth of the ordered carbon-based nanotip structures are considered. The conditions allowing one to predict the size of particles that can pass through the plasma sheath and softly land onto the surface are formulated. The possibility of soft nano-cluster deposition without any additional acceleration common for some existing nano-cluster deposition schemes is demonstrated. The effect of the substrate heating power and the average atomic mass of neutral species is studied numerically and verified experimentally.

Dynamics deflection analysis of nanoelectromechanical switches based on carbon nanotubes

This paper aims to develop an effective numerical simulation technique for the dynamic deflection analysis of nanotubes-based nanoswitches. The nanoswitch is simplified to a continuum structure, and some key material parameters are extracted from typical molecular dynamics (MD). An advanced local meshless formulation is applied to obtain the discretized dynamic equations for the numerical solution. The developed numerical technique is firstly validated by the static deflection analyses of nanoswitches, and then, the fundamental dynamic properties of nanoswitches are analyzed. A parametric comparison with the results in the literature and from experiments shows that the developed modelling approach is accurate, efficient and effective.

Comparing charge transport predictions for a ternary electrolyte using the Maxwell–Stefan and Nernst–Planck equations

In this work, we investigate and compare the Maxwell–Stefan and Nernst–Planck equations for modeling multicomponent charge transport in liquid electrolytes. Specifically, we consider charge transport in the Li+/I−/I3−/ACN ternary electrolyte originally found in dye-sensitized solar cells. We employ molecular dynamics simulations to obtain the Maxwell–Stefan diffusivities for this electrolyte. These simulated diffusion coefficients are used in a multicomponent charge transport model based on the Maxwell– Stefan equations, and this is compared to a Nernst–Planck based model which employs binary diffusion coefficients sourced from the literature. We show that significant differences between the electrolyte concentrations at electrode interfaces, as predicted by the Maxwell–Stefan and Nernst–Planck models, can occur. We find that these differences are driven by a pressure term that appears in the Maxwell–Stefan equations. We also investigate what effects the Maxwell–Stefan diffusivities have on the simulated charge transport. By incorporating binary diffusivities found in the literature into the Maxwell–Stefan framework, we show that the simulated transient concentration profiles depend on the diffusivities; however, the simulated equilibrium profiles remain unaffected.

Molecular dynamics study of dynamic buckling properties of nanowires with defects

Based on the embedded atom method (EAM) and molecular dynamics (MD) method, the deformation properties of Cu nanowires with different single defects under dynamic compression have been studied. The mechanical behaviours of the perfect nanowire are first studied, and the critical stress decreases with the increase of the nanowire’s length, which is well agreed with the modified Euler theory. We then consider the effects to the buckling phenomenon resulted from different defects. It is found that obvious decrease of the critical stress is resulted from different defects, and the largest decrease is found in nanowire with the surface vertical defect. Surface defects are found exerting larger influence than internal defects. The buckling duration is found shortened due to different defects except the nanowire with surface horizon defect, which is also found possessing the largest deflection. Different deflections are also observed for different defected nanowires. It is find that due to surface defects, only deflection in one direction is happened, but for internal defects, more complex deflection circumstances are observed.

Krylov subspace approximations for the exponential Euler method: Error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product φ(tA)b where φ(z) = [exp(z) - 1]/z. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where A is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to φ(tA)b, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error.

Compensation of unmodeled aircraft dynamics in airborne inspection of linear infrastructure assets

Fixed-wing aircraft equipped with downward pointing cameras and/or LiDAR can be used for inspecting approximately piecewise linear assets such as oil-gas pipelines, roads and power-lines. Automatic control of such aircraft is important from a productivity and safety point of view (long periods of precision manual flight at low-altitude is not considered reasonable from a safety perspective). This paper investigates the effect of any unwanted coupling between guidance and autopilot loops (typically caused by unmodeled delays in the aircraft’s response), and the specific impact of any unwanted dynamics on the performance of aircraft undertaking inspection of piecewise linear corridor assets (such as powerlines). Simulation studies and experimental flight tests are used to demonstrate the benefits of a simple compensator in mitigating the unwanted lateral oscillatory behaviour (or coupling) that is caused by unmodeled time constants in the aircraft dynamics.

A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

The Wright-Fisher model is an Itô stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].

Stable strong order 1.0 schemes for solving stochastic ordinary differential equations

The Balanced method was introduced as a class of quasi-implicit methods, based upon the Euler-Maruyama scheme, for solving stiff stochastic differential equations. We extend the Balanced method to introduce a class of stable strong order 1. 0 numerical schemes for solving stochastic ordinary differential equations. We derive convergence results for this class of numerical schemes. We illustrate the asymptotic stability of this class of schemes is illustrated and is compared with contemporary schemes of strong order 1. 0. We present some evidence on parametric selection with respect to minimising the error convergence terms. Furthermore we provide a convergence result for general Balanced style schemes of higher orders.

Pulse dynamics in a three-component system : existence analysis

In this article, we analyze the three-component reaction-diffusion system originally developed by Schenk et al. (PRL 78:3781–3784, 1997). The system consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a prototype three-component system that generates rich pulse dynamics and interactions, and this richness is the main motivation for the analysis we present. We demonstrate the existence of stationary one-pulse and two-pulse solutions, and travelling one-pulse solutions, on the real line, and we determine the parameter regimes in which they exist. Also, for one-pulse solutions, we analyze various bifurcations, including the saddle-node bifurcation in which they are created, as well as the bifurcation from a stationary to a travelling pulse, which we show can be either subcritical or supercritical. For two-pulse solutions, we show that the third component is essential, since the reduced bistable two-component system does not support them. We also analyze the saddle-node bifurcation in which two-pulse solutions are created. The analytical method used to construct all of these pulse solutions is geometric singular perturbation theory, which allows us to show that these solutions lie in the transverse intersections of invariant manifolds in the phase space of the associated six-dimensional travelling wave system. Finally, as we illustrate with numerical simulations, these solutions form the backbone of the rich pulse dynamics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among others.

Simulation of phosphine flow in vertical grain storage : a preliminary numerical study

To fumigate grain stored in a silo, phosphine gas is distributed by a combination of diffusion and fan-forced advection. This initial study of the problem mainly focuses on the advection, numerically modelled as fluid flow in a porous medium. We find satisfactory agreement between the flow predictions of two Computational Fluid Dynamics packages, Comsol and Fluent. The flow predictions demonstrate that the highest velocity (>0.1 m/s) occurs less than 0.2m from the inlet and reduces drastically over one metre of silo height, with the flow elsewhere less than 0.002 m/s or 1% of the velocity injection. The flow predictions are examined to identify silo regions where phosphine dosage levels are likely to be too low for effective grain fumigation.

Numerical simulation of stochastic ordinary differential equations in biomathematical modelling

In this work we discuss the effects of white and coloured noise perturbations on the parameters of a mathematical model of bacteriophage infection introduced by Beretta and Kuang in [Math. Biosc. 149 (1998) 57]. We numerically simulate the strong solutions of the resulting systems of stochastic ordinary differential equations (SDEs), with respect to the global error, by means of numerical methods of both Euler-Taylor expansion and stochastic Runge-Kutta type.

Multidisciplinary design and flight testing of a remote gas/particle airborne sensor system

The main objective of this paper is to describe the development of a remote sensing airborne air sampling system for Unmanned Aerial Systems (UAS) and provide the capability for the detection of particle and gas concentrations in real time over remote locations. The design of the air sampling methodology started by defining system architecture, and then by selecting and integrating each subsystem. A multifunctional air sampling instrument, with capability for simultaneous measurement of particle and gas concentrations was modified and integrated with ARCAA’s Flamingo UAS platform and communications protocols. As result of the integration process, a system capable of both real time geo-location monitoring and indexed-link sampling was obtained. Wind tunnel tests were conducted in order to evaluate the performance of the air sampling instrument in controlled nonstationary conditions at the typical operational velocities of the UAS platform. Once the remote fully operative air sampling system was obtained, the problem of mission design was analyzed through the simulation of different scenarios. Furthermore, flight tests of the complete air sampling system were then conducted to check the dynamic characteristics of the UAS with the air sampling system and to prove its capability to perform an air sampling mission following a specific flight path.