Parameter estimation in approximate enzyme kinetics models

Enzyme kinetic modeling is based on a very specific approximation to a more detailed model; this approximation is often called the Michaelis-Menten approximation. Its success lies not only in an initial concentration difference that is set experimentally, but also in a certain difference among kinetic rate constants. Other differences among these constants lead to additional model approximations, often much simpler than the Michaelis-Menten approximation. Under the conditions where these alternate approximations apply, the Michaelis-Menten approximation would give misleading parameter estimates.     

A Modification of the Edgeworth Approximation in the AR (1) Model

This paper considers the sampling distribution problem of the least squares estimator for the parameter of some special autoregressive models. The Edgeworth approximation has been derived and a modification is proposed to improve its accuracy. Comparisons with the exact distribution and the so called Edgeworth-B approximation have been discussed. The results show that the proposed approximation performs more accurately than the Edgeworth-B approximation, especially, when models are close to the non-stationary boundary.     

Semiclassical approximations of stochastic epidemiological processes towards parameter estimation using as prime example the SIS system with import

In this paper we investigate several schemes to approximate the stationary distribution of the stochastic SIS system with import. We begin by presenting the model and analytically computing its stationary distribution. We then approximate this distribution using Kramers–Moyal approximation, van Kampen's system size expansion, and a semiclassical scheme, also called WKB or eikonal approximation depending on its different applications in physics. For the semiclassical scheme, done in the context of the Hamilton–Jacobi formalism, two approaches are taken. In the first approach we assume a semiclassical ansatz for the generating function, while in the second the solution of the master equation is approximated directly. The different schemes are compared and the semiclassical approximation, which performs better, is then used to analyse the time dependent solution of stochastic systems for which no analytical expression is known. Stochastic epidemiological models are studied in order to investigate how far such semiclassical approximations can be used for parameter estimation.     

Speed of invasion in lattice population models: pair-edge approximation

 We propose a simple approach to approximating the speed of invasion in lattice population models. Approximate critical parameter values for successful invasion are then found by solving for zero wave speed. The approximation is based on describing the occupied region by the ordinary pair approximation, and using quasi-steady-state pair approximations to describe the leading edge of the wave front. We illustrate this idea using the basic contact process on the 1 and 2 dimensional lattice (with and without nearest-neighbor migration), finding very good agreement between the approximation and simulation results. The approximate critical values obtained by our approximation are significantly more accurate than those obtained by the ordinary pair approximation. Received 4 September 1996     

Bayesian information criterion for censored survival models

We investigate the Bayesian Information Criterion (BIC) for variable selection in models for censored survival data. Kass and Wasserman (1995, Journal of the American Statistical Association 90, 928-934) showed that BIC provides a close approximation to the Bayes factor when a unit-information prior on the parameter space is used. We propose a revision of the penalty term in BIC so that it is defined in terms of the number of uncensored events instead of the number of observations. For a simple censored data model, this revision results in a better approximation to the exact Bayes factor based on a conjugate unit-information prior. In the Cox proportional hazards regression model, we propose defining BIC in terms of the maximized partial likelihood. Using the number of deaths rather than the number of individuals in the BIC penalty term corresponds to a more realistic prior on the parameter space and is shown to improve predictive performance for assessing stroke risk in the Cardiovascular Health Study.     

Comparing approximations to spatio-temporal models for epidemics with local spread

Analytical methods for predicting and exploring the dynamics of stochastic, spatially interacting populations have proven to have useful application in epidemiology and ecology. An important development has been the increasing interest in spatially explicit models, which require more advanced analytical techniques than the usual mean-field or mass-action approaches. The general principle is the derivation of differential equations describing the evolution of the expected population size and other statistics. As a result of spatial interactions no closed set of equations is obtained. Nevertheless, approximate solutions are possible using closure relations for truncation. Here we review and report recent progress on closure approximations applicable to lattice models with nearest-neighbour interactions, including cluster approximations and elaborations on the pair (or pairwise) approximation. This study is made in the context of an SIS model for plant-disease epidemics introduced in Filipe and Gibson (1998, Studying and approximating spatio-temporal models for epidemic spread and control, Phil. Trans. R. Soc. Lond. B 353, 2153–2162) of which the contact process [Harris, T. E. (1974), Contact interactions on a lattice, Ann. Prob. 2, 969] is a special case. The various methods of approximation are derived and explained and their predictions are compared and tested against simulation. The merits and limitations of the various approximations are discussed. A hybrid pairwise approximation is shown to provide the best predictions of transient and long-term, stationary behaviour over the whole parameter range of the model.     

Numerical discretization-based estimation methods for ordinary differential equation models via penalized spline smoothing with applications in biomedical research

Differential equations are extensively used for modeling dynamics of physical processes in many scientific fields such as engineering, physics, and biomedical sciences. Parameter estimation of differential equation models is a challenging problem because of high computational cost and high-dimensional parameter space. In this article, we propose a novel class of methods for estimating parameters in ordinary differential equation (ODE) models, which is motivated by HIV dynamics modeling. The new methods exploit the form of numerical discretization algorithms for an ODE solver to formulate estimating equations. First, a penalized-spline approach is employed to estimate the state variables and the estimated state variables are then plugged in a discretization formula of an ODE solver to obtain the ODE parameter estimates via a regression approach. We consider three different order of discretization methods, Euler's method, trapezoidal rule, and Runge-Kutta method. A higher-order numerical algorithm reduces numerical error in the approximation of the derivative, which produces a more accurate estimate, but its computational cost is higher. To balance the computational cost and estimation accuracy, we demonstrate, via simulation studies, that the trapezoidal discretization-based estimate is the best and is recommended for practical use. The asymptotic properties for the proposed numerical discretization-based estimators are established. Comparisons between the proposed methods and existing methods show a clear benefit of the proposed methods in regards to the trade-off between computational cost and estimation accuracy. We apply the proposed methods t an HIV study to further illustrate the usefulness of the proposed approaches.     

Novel moment closure approximations in stochastic epidemics

Moment closure approximations are used to provide analytic approximations to non-linear stochastic population models. They often provide insights into model behaviour and help validate simulation results. However, existing closure schemes typically fail in situations where the population distribution is highly skewed or extinctions occur. In this study we address these problems by introducing novel second-and third-order moment closure approximations which we apply to the stochastic SI and SIS epidemic models. In the case of the SI model, which has a highly skewed distribution of infection, we develop a second-order approximation based on the beta-binomial distribution. In addition, a closure approximation based on mixture distribution is developed in order to capture the behaviour of the stochastic SIS model around the threshold between persistence and extinction. This mixture approximation comprises a probability distribution designed to capture the quasi-equilibrium probabilities of the system and a probability mass at 0 which represents the probability of extinction. Two third-order versions of this mixture approximation are considered in which the log-normal and the beta-binomial are used to model the quasi-equilibrium distribution. Comparison with simulation results shows: (1) the beta-binomial approximation is flexible in shape and matches the skewness predicted by simulation as shown by the stochastic SI model and (2) mixture approximations are able to predict transient and extinction behaviour as shown by the stochastic SIS model, in marked contrast with existing approaches. We also apply our mixture approximation to approximate a likehood function and carry out point and interval parameter estimation.     

Reproduction numbers for epidemics on networks using pair approximation

One way to describe the spread of an infection on a network is by using the method of pair approximation. This method is a deterministic pair-based variant of the usual methods used to describe the progress of an epidemic in randomly mixing populations. However, although the ideas of pair approximation are intuitively clear, it is not straightforward to make all assumptions used explicit. Furthermore, in literature problems arise in defining basic quantities like the basic reproduction number R(0) and the real-time epidemic growth rate parameter r. We formulate the pair approximations and the needed assumptions explicitly. We discuss problems inherent to this method. Furthermore, we define a new reproduction number, similar to R(0) and a new real-time growth rate parameter similar to r. We illustrate the methods of the paper by an example for which we can compare the approximation of the reproduction number with exact results.     

A stochastic model for a single click of Muller's ratchet

This work presents a new approach to Muller's ratchet, where Haigh's model is approximately mapped into a simpler model that describes the behaviour of a population after a click of the ratchet, i.e., after loss of what was the fittest class. This new model predicts the distribution of times to the next click of the ratchet and is equivalent to a Wright-Fisher model for a population of haploid asexual individuals with one locus and two alleles. Within this model, the fittest members of a population correspond to carriers of one allele, while all other individuals have suboptimal fitness and are represented as carriers of the other allele. In this way, all suboptimal fitness individuals are amalgamated into a single “mutant” class.The approach presented here has some limitations and the potential for improvement. However, it does lead to results for the rate of the ratchet that, over a wide range of parameters, are accurate within one order of magnitude of simulation results. This contrasts with existing approaches, which are designed for only one or other of the two different parameter regimes known for the ratchet and are more accurate only in the parameter regime they were designed for.Numerical results are presented for the mean time between clicks of the ratchet for (i) the Wright-Fisher model, (ii) a diffusion approximation of this model and (iii) individually based simulations of a full model. The diffusion approximation is validated over a wide range of parameters by its close agreement with the Wright-Fisher model.The present work predicts that: (a) the time between clicks of the ratchet is insensitive to the value of the selection coefficient when the genomic mutation rate is large compared with the selection coefficient against a deleterious mutation, (b) the time interval between clicks of the ratchet has, approximately, an exponential distribution (or its discrete analogue). It is thus possible to determine the variance in times between clicks, given the expected time between clicks. Evidence for both (a) and (b) is seen in simulations.     

An online self-organizing scheme for Parsimonious and Accurate Fuzzy Neural Networks

In this paper, an online self-organizing scheme for Parsimonious and Accurate Fuzzy Neural Networks (PAFNN), and a novel structure learning algorithm incorporating a pruning strategy into novel growth criteria are presented. The proposed growing procedure without pruning not only simplifies the online learning process but also facilitates the formation of a more parsimonious fuzzy neural network. By virtue of optimal parameter identification, high performance and accuracy can be obtained. The learning phase of the PAFNN involves two stages, namely structure learning and parameter learning. In structure learning, the PAFNN starts with no hidden neurons and parsimoniously generates new hidden units according to the proposed growth criteria as learning proceeds. In parameter learning, parameters in premises and consequents of fuzzy rules, regardless of whether they are newly created or already in existence, are updated by the extended Kalman filter (EKF) method and the linear least squares (LLS) algorithm, respectively. This parameter adjustment paradigm enables optimization of parameters in each learning epoch so that high performance can be achieved. The effectiveness and superiority of the PAFNN paradigm are demonstrated by comparing the proposed method with state-of-the-art methods. Simulation results on various benchmark problems in the areas of function approximation, nonlinear dynamic system identification and chaotic time-series prediction demonstrate that the proposed PAFNN algorithm can achieve more parsimonious network structure, higher approximation accuracy and better generalization simultaneously.     

Robustness of the approximate likelihood of the protracted speciation model

The protracted speciation model presents a realistic and parsimonious explanation for the observed slowdown in lineage accumulation through time, by accounting for the fact that speciation takes time. A method to compute the likelihood for this model given a phylogeny is available and allows estimation of its parameters (rate of initiation of speciation, rate of completion of speciation and extinction rate) and statistical comparison of this model to other proposed models of diversification. However, this likelihood computation method makes an approximation of the protracted speciation model to be mathematically tractable: it sometimes counts fewer species than one would do from a biological perspective. This approximation may have large consequences for likelihood‐based inferences: it may render any conclusions based on this method completely irrelevant. Here, we study to what extent this approximation affects parameter estimations. We simulated phylogenies from which we reconstructed the tree of extant species according to the original, biologically meaningful protracted speciation model and according to the approximation. We then compared the resulting parameter estimates. We found that the differences were larger for high values of extinction rates and small values of speciation‐completion rates. Indeed, a long speciation‐completion time and a high extinction rate promote the appearance of cases to which the approximation applies. However, surprisingly, the deviation introduced is largely negligible over the parameter space explored, suggesting that this approximate likelihood can be applied reliably in practice to estimate biologically relevant parameters under the original protracted speciation model.     

Research on the parameter inversion problem of prestack seismic data based on improved differential evolution algorithm

The parameter inversion technology composed by intelligent algorithm and AVO inversion for prestack seismic data provides a comparatively effective identification method for oil-gas exploration. However, traditionally intelligent iterative algorithm, such as, genetic algorithm, shows many disadvantages in solving this problem, including highly depending on initial model, fast convergence in algorithm and being easy to fall into local optimal. Therefore, an unsatisfied inversion performance is produced. In order to solve the above problems, this paper proposes a parameter inversion method based on improved differential evolution algorithm which is better in solving parameter inversion problems of prestack seismic data. In the proposed algorithm, aims at the Aki and Rechard approximation formula used specific initialization strategy, then the initialization parameter curve more smooth. Otherwise, the new algorithm has many advantages, such as, fast computing speed, simple operation, a low independence to initial model and good global convergence, this algorithm is the right choice in solving the parameter inversion problem based on pre-stack seismic data of real number encoding.     

Shape, Size, and Robustness: Feasible Regions in the Parameter Space of Biochemical Networks

The concept of robustness of regulatory networks has received much attention in the last decade. One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional. In this paper, we show that, in addition to volume, the geometry of this region has important consequences for the robustness and the fragility of a network. We develop an approximation within which we could algebraically specify the feasible region. We analyze the segment polarity gene network to illustrate our approach. The study of random walks in the parameter space and how they exit the feasible region provide us with a rich perspective on the different modes of failure of this network model. In particular, we found that, between two alternative ways of activating Wingless, one is more robust than the other. Our method provides a more complete measure of robustness to parameter variation. As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks.     

Unsteady wall shear stress in a distensible tube

An asymptotic expression of the wall shear stress (WSS) in an elastic tube is deduced for small values of the Womersley parameter. In the case of a rigid tube this asymptotic expression is shown to compare better with the exact solution than Poiseuille's or Lambossy's approximations. Its integration in a one-dimensional model of the internal carotid artery blood flow predicts more marked systolic and less marked diastolic WSS than those predicted by the commonly used Poiseuille's approximation.     

A new look at the critical community size for childhood infections

Quasi-stationarity and time to extinction are studied for the classic endemic model. Attention is restricted to the transition region in parameter space where the quasi-stationary distribution is non-normal. A new approximation of the marginal distribution of infected individuals in quasi-stationarity is presented. It leads to a simple explicit expression for an approximation of the critical community size in terms of model parameters.     

A simple predictive density based on the p*-formula

The predictive density proposed by Harris (1989) is based onintegrating the density for a new observation with respect tothe estimated sampling distribution of the maximum likelihoodestimator of the unknown parameter. This has good properties,but is rather complicated to compute even for simple models.An approximation to the Harris proposal is considered whichconsists of approximating the sampling distribution of the maximumlikelihood estimator of the unknown parameter by Barndorff-Nielsen's(1983) p*-formula, and then using a Laplace approximation withO(n^–1) correction terms for integrating out the parameter.The result can generally be expressed in terms of standard likelihoodderivatives, and takes a quite simple form for exponential familiesand for location models.     

Variable selection and parameter estimation of viral amplification in vero cell cultures dedicated to the production of a dengue vaccine

In this study, a dynamic model of a Vero cell culture-based dengue vaccine production process is developed. The approach consists in describing the process dynamics as functions of the whole living (uninfected and infected) biomass whereas previous works are based on population balance approaches. Based on the assumption that infected biomass evolves faster than other variable, the model can be simplified using a slow-fast approximation. The structural identifiability of the model is analysed using differential algebra as implemented in the software DAISY. The model parameters are inferred from experimental datasets collected from an actual vaccine production process and the model predictive capability is confirmed both in direct and cross-validation. The model prediction shows the impact of the metabolism on virus yield and confirms observations reported in previous studies. Multi-modality and sensitivity analysis complement the parameter estimation, and allow to obtain confidence intervals on both parameters and state estimates. Finally, the model is used to compute the maximum infectious virus yield that can be obtained for different combinations of multiplicity of infection (MOI) and time of infection (TOI). © 2018 American Institute of Chemical Engineers Biotechnol. Prog., 35: e2687, 2019