Statistical inference from single channel records: two-state Markov model with limited time resolution

Though stochastic models are widely used to describe single ion channel behaviour, statistical inference based on them has received little consideration. This paper describes techniques of statistical inference, in particular likelihood methods, suitable for Markov models incorporating limited time resolution by means of a discrete detection limit. To simplify the analysis, attention is restricted to two-state models, although the methods have more general applicability. Non-uniqueness of the mean open-time and mean closed-time estimators obtained by moment methods based on single exponential approximations to the apparent open-time and apparent closed-time distributions has been reported. The present study clarifies and extends this previous work by proving that, for such approximations, the likelihood equations as well as the moment equations (usually) have multiple solutions. Such non-uniqueness corresponds to non-identifiability of the statistical model for the apparent quantities. By contrast, higher-order approximations yield theoretically identifiable models. Likelihood-based estimation procedures are developed for both single exponential and bi-exponential approximations. The methods and results are illustrated by numerical examples based on literature and simulated data, with consideration given to empirical distributions and model control, likelihood plots, and point estimation and confidence regions.     

Application of Petri net based analysis techniques to signal transduction pathways

Background  

Signal transduction pathways are usually modelled using classical quantitative methods, which are based on ordinary differential equations (ODEs). However, some difficulties are inherent in this approach. On the one hand, the kinetic parameters involved are often unknown and have to be estimated. With increasing size and complexity of signal transduction pathways, the estimation of missing kinetic data is not possible. On the other hand, ODEs based models do not support any explicit insights into possible (signal-) flows within the network. Moreover, a huge amount of qualitative data is available due to high-throughput techniques. In order to get information on the systems behaviour, qualitative analysis techniques have been developed. Applications of the known qualitative analysis methods concern mainly metabolic networks. Petri net theory provides a variety of established analysis techniques, which are also applicable to signal transduction models. In this context special properties have to be considered and new dedicated techniques have to be designed.     

Analysis and modeling of suspended solids from high-frequency monitoring in a stormwater treatment wetland

Tools for modeling pulsed flows and constituent fluxes in wetlands, although well developed in theory, have not been well tested in practice. High-frequency monitoring of suspended solids and flows in a stormwater treatment wetland enabled application and analysis of these tools. A dynamic flow- and volume-weighted time variable, analogous to the retention time in steady-flow systems, is one important tool tested in this study. Cross-correlations with time lags demonstrated that the dynamic time variable was a better predictive variable of pulsed events than was the standard, static time variable. Although plug-flow models are typically used for steady-flow wetlands, residence time distribution (RTD) models are indispensable for describing pulsed flows and constituent fluxes in wetlands. This study demonstrated that RTD modeling with reaction kinetics of suspended solids during storm events produces a better explanation of outflow data than possible with steady, plug-flow models. Using only input and output data, an RTD model explained sedimentation rates with less unexplained variance than the standard, plug-flow model. The results of this study underscore the importance and utility of RTD modeling for complex flows.     

Solution of functional difference equations from behavioral theory

Behavioral models based on Markovian decision processes lead to functional difference equations for quantities such as the mean lifetime of the forager and the probability of reproductive success of the forager. In this paper, asymptotic and iterative methods are developed for the solution of such equations. The asymptotic methods are compared with numerical simulations. The iterative methods can be proved by a simple application of contraction mapping theorems.     

Current status and challenges in connecting models of erythrocyte metabolism to experimental reality

Detailed kinetic models of human erythrocyte metabolism have served to summarize the vast literature and to predict outcomes from laboratory and “Nature's” experiments on this simple cell. Mathematical methods for handling the large array of nonlinear ordinary differential equations that describe the time dependence of this system are well developed, but experimental methods that can guide the evolution of the models are in short supply. NMR spectroscopy is one method that is non-selective with respect to analyte detection but is highly specific with respect to their identification and quantification. Thus time courses of metabolism are readily recorded for easily changed experimental conditions. While the data can be simulated, the systems of equations are too complex to allow solutions of the inverse problem, namely parameter-value estimation for the large number of enzyme and membrane-transport reactions operating in situ as opposed to in vitro. Other complications with the modelling include the dependence of cell volume on time, and the rates of membrane transport processes are often dependent on the membrane potential. These matters are discussed in the light of new modelling strategies.     

The frequency response of electrochemical wall shear probes in pulsatile flow

The frequency response of surface-mounted electrochemical mass transfer probes used to deduce wall shear rates has been investigated experimentally for the case of fully developed laminar pulsatile flow in a straight tube. Generally good agreement is found with the asymptotic results obtained by Lighthill's methods. The significance of the results with regard to the investigation of models of pulsatile flows of physiological interest is discussed. It is concluded that the frequency-dependent phase and amplitude corrections required to obtain accurate wall shear measurements are of such magnitudes as to render impractical the use of electrochemical probes to determine wall shear rates in these flows.     

Equations for measuring the rate of DNA chain growth and replicon size by density labeling techniques

The effects of initiation, termination and clustering of replicons have been considered in the development of equations which are used to measure DNA chain elongation in mammalian cells by density gradient techniques. These equations can be used to determine the average replicon size as well as the rate of chain elongation. Experimental data from WI-38 and HeLa cells are presented which show the applicability of these methods. Several models of replication initiation and termination have been developed. Some of these models can be eliminated experimentally and the experimental results are consistent with the notion that neighboring replicons are initiated simultaneously or sequentially with short (～-0·5 min) interinitiation time.     

Rapid development of indirect effects in ecological networks

Indirect effects are important components of ecological and evolutionary interactions that may maintain biodiversity, enable or inhibit invasive species, and challenge ecosystem assessment and management. A central hypothesis of Network Environ Analysis (NEA), one type of ecological network analysis, is that indirect flows tend to dominate direct flows in ecosystem networks of conservative substance exchanges. However, current NEA methods assume that these ecosystems are stationary (i.e. time invariant exchange rates), which is unlikely to be true for many ecosystems for interesting time and space scales. For the work reported here, we investigated the sensitivity of the dominance of indirect effects hypothesis to the stationary modeling assumption by determining the development rate of indirect effects and flow intensity, as expressed as the number of transfer steps, in thirty‐one ecosystem models. We hypothesized that indirect effects develop rapidly in ecological networks, but that they would develop faster in biogeochemically based models than in trophically based models. In contrast, our results show that indirect effects develop rapidly in all thirty‐one models examined. In 94% of the models, indirect flows exceeded direct flows by a pathway length of 3. This indicates that ecological systems do not need to maintain a particular configuration for long for indirect effects to dominate. Thus, the dominance of indirect effects hypothesis remains plausible. We also found that biogeochemical models tended to require more of the extended path network than the trophic models to account for 50% and 95% of the total system activity, but that both types of models required more of the power series than is typically considered in engineered systems. These results succinctly illustrate the complexity of ecological systems and help explain why they are challenging to assess and manage.     

Analysis of various indefinite self-associations of the AK type

Although we and others have developed equations to analyze for some indefinite self-associations that might be encountered, it is felt in some cases that these models, known as the sequential, equal equilibrium constant (SEK) models, might overestimate the size of aggregates encountered at higher solute concentrations. Thus, Garland and Christian proposed two attenuated equilibrium constant (AK) models that might overcome this problem. Their methods were restricted to ideal solutions and to osmometric procedures. We have removed these restrictions, and we have developed equations for analyzing four AK models that might be encountered. Various tests to aid in distinguishing these models are presented. These procedures have been tested with two simulated examples of a Type III AK indefinite self-association.     

Empirical Dynamic Material Flow Model for Tungsten in the USA

Multivariate polynomial regression (MPR) analysis was implemented to develop a nonlinear dynamic material flow model (DMFM) of tungsten in the United States for the years 1975–2000 without assumptions for lifetime distributions within reservoirs. Two external economic factors, the Consumer Price Index and the Industrial Production Index, were included as possible exogenous variables. Six types of vector time‐series models were developed using multilinear, simple interaction, and MPR models, each with and without the exogenous economic variables. The DMFMs developed in this work make one‐step‐ahead predictions. That is, the material flows in a given year were predicted using flows and exogenous variables from previous years. In contrast to approaches that utilize assumed lifetime distributions for material within reservoirs, such as the Weibull distribution, the approach used here is completely data driven. MPR models produced statistically better results than linear models for all 13 flows that were modeled. Four of these models used simple interaction terms (which we call linear interaction terms), and two of these incorporated exogenous variables. The other nine models utilized higher‐degree terms with interactions (called multivariate polynomial terms), and two of these incorporated exogenous variables. We conclude that nonlinear vector time series are capable of identifying complex relationships among material flows and exogenous variables. An understanding of these relationships has potential for managing, conserving, and/or forecasting the use of a resource.     

A variational calculus approach to the suspension flow problem using the minimal force hypothesis

Analytic expressions for the velocity profile and distribution of neutrally buoyant particles in laminar flow were obtained as functions of the radial distance. A modified Einstein viscosity model and the hypothesis that the total force on all the particles flowing in the tube is a minimum were used. The methods of the variational calculus were used in the mathematical development. A velocity profile differing only slightly from the parabolic form of that for Hagan-Poiseuille flow was obtained. For particle distribution the equations developed predict a maximum concentration along the center-line for some flows and a maximum concentration in a ring some distance from the center line in other flows. Both of these concentration profiles have been observed experimentally. Quantitative predictions from the equations derived must await further experimental work to permit calculation of the parameters included in the equations.     

The Path Integral Formulation of Climate Dynamics

The chaotic nature of the atmospheric dynamics has stimulated the applications of methods and ideas derived from statistical dynamics. For instance, ensemble systems are used to make weather predictions recently extensive, which are designed to sample the phase space around the initial condition. Such an approach has been shown to improve substantially the usefulness of the forecasts since it allows forecasters to issue probabilistic forecasts. These works have modified the dominant paradigm of the interpretation of the evolution of atmospheric flows (and oceanic motions to some extent) attributing more importance to the probability distribution of the variables of interest rather than to a single representation. The ensemble experiments can be considered as crude attempts to estimate the evolution of the probability distribution of the climate variables, which turn out to be the only physical quantity relevant to practice. However, little work has been done on a direct modeling of the probability evolution itself. In this paper it is shown that it is possible to write the evolution of the probability distribution as a functional integral of the same kind introduced by Feynman in quantum mechanics, using some of the methods and results developed in statistical physics. The approach allows obtaining a formal solution to the Fokker-Planck equation corresponding to the Langevin-like equation of motion with noise. The method is very general and provides a framework generalizable to red noise, as well as to delaying differential equations, and even field equations, i.e., partial differential equations with noise, for example, general circulation models with noise. These concepts will be applied to an example taken from a simple ENSO model.     

Dispersal and competition models for plants

New models for seed dispersal and competition between plant species are formulated and analyzed. The models are integrodifference equations, discrete in time and continuous in space, and have applications to annual and perennial species. The spread or invasion of a single plant species into a geographic region is investigated by studying the travelling wave solutions of these equations. Travelling wave solutions are shown to exist in the single-species models and are compared numerically. The asymptotic wave speed is calculated for various parameter values. The single-species integrodifference equations are extended to a model for two competing annual plants. Competition in the two-species model is based on a difference equation model developed by Pakes and Maller [26]. The two-species model with competition and dispersal yields a system of integrodifference equations. The effects of competition on the travelling wave solutions of invading plant species is investigated numerically.     

Biofuel production improvement with genome-scale models: The role of cell composition

Genome-scale models have developed into a vital tool for rational metabolic engineering. These models balance cofactors and energetic requirements and determine biosynthetic precursor availability in response to environmental and genetic perturbations. In particular, allocation of additional reducing power is an important strategy for engineering potential biofuel production from microbes. Many potential biofuel solvents induce biomolecular changes on the host organism that are not yet captured by genome-scale models. Here, methods of construction for several biomass constituting equations are reviewed along with potential changes to cellular composition with potential biofuels exposure. The biomass constituting equations of potential host organisms with existing genome-scale models are compared side-by-side to explore their evolution over the years and to explore differences that arise when these equations are compiled by different research groups. Genome-scale model simulation results attempt to address and provide guidance for further research into: (i) whether inconsistencies in the biomass constituting equations are relevant to predictions of solvent production, (ii) what level of detail is necessary to accurately describe cellular composition, and (iii) future developments that may enable more accurate characterizations of biomolecular composition.     

Automatic noninvasive measurement of systolic blood pressure using photoplethysmography

Background

Systolic blood flow has been simulated in the abdominal aorta and the superior mesenteric artery. The simulations were carried out using two different computational hemodynamic methods: the finite element method to solve the Navier Stokes equations and the lattice Boltzmann method.

Results

We have validated the lattice Boltzmann method for systolic flows by comparing the veloCity and pressure profiles of simulated blood flow between methods. We have also analyzed flow-specific characteristics such as the formation of a vortex at curvatures and traces of flow.

Conclusion

The lattice Boltzmann Method is as accurate as a Navier Stokes solver for computing complex blood flows. As such it is a good alternative for computational hemodynamics, certainly in situation where coupling to other models is required.     

Geometric and potential driving formation and evolution of biomolecular surfaces

This paper presents new geometrical flow equations for the theoretical modeling of biomolecular surfaces in the context of multiscale implicit solvent models. To account for the local variations near the biomolecular surfaces due to interactions between solvent molecules, and between solvent and solute molecules, we propose potential driven geometric flows, which balance the intrinsic geometric forces that would occur for a surface separating two homogeneous materials with the potential forces induced by the atomic interactions. Stochastic geometric flows are introduced to account for the random fluctuation and dissipation in density and pressure near the solvent–solute interface. Physical properties, such as free energy minimization (area decreasing) and incompressibility (volume preserving), are realized by some of our geometric flow equations. The proposed approach for geometric and potential forces driving the formation and evolution of biological surfaces is illustrated by extensive numerical experiments and compared with established minimal molecular surfaces and molecular surfaces. Local modification of biomolecular surfaces is demonstrated with potential driven geometric flows. High order geometric flows are also considered and tested in the present work for surface generation. Biomolecular surfaces generated by these approaches are typically free of geometric singularities. As the speed of surface generation is crucial to implicit solvent model based molecular dynamics, four numerical algorithms, a semi-implicit scheme, a Crank–Nicolson scheme, and two alternating direction implicit (ADI) schemes, are constructed and tested. Being either stable or conditionally stable but admitting a large critical time step size, these schemes overcome the stability constraint of the earlier forward Euler scheme. Aided with the Thomas algorithm, one of the ADI schemes is found to be very efficient as it balances the speed and accuracy.      

Semiparametric Transformation Models with Time‐Varying Coefficients for Recurrent and Terminal Events

Summary In this article, we propose a family of semiparametric transformation models with time‐varying coefficients for recurrent event data in the presence of a terminal event such as death. The new model offers great flexibility in formulating the effects of covariates on the mean functions of the recurrent events among survivors at a given time. For the inference on the proposed models, a class of estimating equations is developed and asymptotic properties of the resulting estimators are established. In addition, a lack‐of‐fit test is provided for assessing the adequacy of the model, and some tests are presented for investigating whether or not covariate effects vary with time. The finite‐sample behavior of the proposed methods is examined through Monte Carlo simulation studies, and an application to a bladder cancer study is also illustrated.     

Magnetic resonance microscopy analysis of advective transport in a biofilm reactor

In this article we present magnetic resonance microscopy (MRM) characterization of the advective transport in a biofilm capillary reactor. The biofilm generates non-axial flows that are up to 20% of the maximum axial velocity. The presence of secondary velocities of this magnitude alters the mass transport in the bioreactor relative to non-biofilm fouled reactors and questions the applicability of empirical mass transfer coefficient approaches. The data are discussed in the context of simulations and models of biofilm transport and conceptual aspects of transport modeling in complex flows are also discussed. The variation in the residence time distribution due to biofilm growth is calculated from the measured propagator of the motion. Dynamical systems methods applied to model fluid mixing in complex flows are indicated as a template for extending mass transport theory to quantitatively incorporate microscale data on the advection field into macroscale mass transfer models.     

An axisymmetric single-path model for gas transport in the conducting airways

In conventional one-dimensional single-path models, radially averaged concentration is calculated as a function of time and longitudinal position in the lungs, and coupled convection and diffusion are accounted for with a dispersion coefficient. The axisymmetric single-path model developed in this paper is a two-dimensional model that incorporates convective-diffusion processes in a more fundamental manner by simultaneously solving the Navier-Stokes and continuity equations with the convection-diffusion equation. A single airway path was represented by a series of straight tube segments interconnected by leaky transition regions that provide for flow loss at the airway bifurcations. As a sample application, the model equations were solved by a finite element method to predict the unsteady state dispersion of an inhaled pulse of inert gas along an airway path having dimensions consistent with Weibel's symmetric airway geometry. Assuming steady, incompressible, and laminar flow, a finite element analysis was used to solve for the axisymmetric pressure, velocity and concentration fields. The dispersion calculated from these numerical solutions exhibited good qualitative agreement with the experimental values, but quantitatively was in error by 20%-30% due to the assumption of axial symmetry and the inability of the model to capture the complex recirculatory flows near bifurcations.