Identification of nonlinear models of biological membranes using the voltage-clamp method

The method for identification of nonlinear systems proposed in 1952 by Hodgkin and Huxley is mathematically justified. A procedure for the application of this method is developed, including the development of the structure of a mathematical model, carrying out a series of tests with special chosen signals, and determination of unknown parameters. Basic requirements for the admissible sets of input and output signals and to the system operator have been determined. It is shown that this operator should be totally continuous and that the minimum number of unknown parameters and the minimum complexity of the operator structure should give an approximation of the necessary quality. The pros and cons of the Hodgkin-Huxley and Noble mathematical models and the methods used for their development are discussed. A structure for the operator for the identification of mathematical models of excitable membranes with a large number of membrane currents is proposed. It is found that the nonlinear electrical properties of biological membranes can be identified using tests with other types of “clamped” parameters, such as the current, ramp voltage, etc.     

Extension of the SAEM algorithm for nonlinear mixed models with 2 levels of random effects

This article focuses on parameter estimation of multilevel nonlinearmixed-effects models (MNLMEMs). These models are used to analyzedata presenting multiple hierarchical levels of grouping (clusterdata, clinical trials with several observation periods, ...).The variability of the individual parameters of the regressionfunction is thus decomposed as a between-subject variabilityand higher levels of variability (e.g. within-subject variability).We propose maximum likelihood estimates of parameters of thoseMNLMEMs with 2 levels of random effects, using an extensionof the stochastic approximation version of expectation–maximization(SAEM)–Monte Carlo Markov chain algorithm. The extendedSAEM algorithm is split into an explicit direct expectation–maximization(EM) algorithm and a stochastic EM part. Compared to the originalalgorithm, additional sufficient statistics have to be approximatedby relying on the conditional distribution of the second levelof random effects. This estimation method is evaluated on pharmacokineticcrossover simulated trials, mimicking theophylline concentrationdata. Results obtained on those data sets with either the SAEMalgorithm or the first-order conditional estimates (FOCE) algorithm(implemented in the nlme function of R software) are compared:biases and root mean square errors of almost all the SAEM estimatesare smaller than the FOCE ones. Finally, we apply the extendedSAEM algorithm to analyze the pharmacokinetic interaction oftenofovir on atazanavir, a novel protease inhibitor, from theAgence Nationale de Recherche sur le Sida 107-Puzzle 2 study.A significant decrease of the area under the curve of atazanaviris found in patients receiving both treatments.     

Parameter estimation in cardiac ionic models

We examine the problem of parameter estimation in mathematical models of excitable cell cardiac electrical activity using the well-known Beeler–Reuter (1977) ionic equations for the ventricular action potential. The estimation problem can be regarded as equivalent to the accurate reconstruction of ionic current kinetics and amplitudes in an excitable cell model, given only action potential experimental data. We show that in the Beeler–Reuter case, all ionic currents may be reasonably reconstructed using an experimental design consisting of action potential recordings perturbed by pseudo-random injection currents.

The Beeler–Reuter model was parameterised into 63 parameters completely defining all membrane current amplitudes and kinetics. Total membrane current was fitted to model-generated experimental data using a ‘data-clamp’ protocol. The experimental data consisted of a default action-potential waveform and an optional series of perturbed waveforms generated by current injections. Local parameter identifiability was ascertained from the reciprocal condition value (1/λ) of the Hessian at the known solution. When fitting to a single action potential waveform, the model was found to be over-determined, having a 1/λ value of 3.6e−14. This value improved slightly to 1.4e−10 when an additional 2 perturbed waveforms were included in the fitting process, suggesting that the additional data did not overly improve the identifiability problem. The additional data, however, did allow the accurate reconstruction of all ionic currents. This indicates that by appropriate experimental design, it may be possible to infer the properties of underlying membrane currents from observation of transmembrane potential waveforms perturbed by pseudo-random currents.     

The precision of results from nonlinear regression analysis of data following a three-parameter one-exponential equation

Simulated experimental data were generated from error-free data following the equation y = A ? Be^?k1 where A, B, and k are constants and were analyzed by iterative nonlinear regression using one of two basic published computer programs. The effect of the simulated experimental error in y on the precision of the computed constants A, B, and k was evaluated. The errors were either independent of y (simple errors) or proportional to y (relative errors) and outliers were sometimes introduced. Other factors investigated were the number of data points per regression, the range of values of y, and the effect of weighting the data. The results show that the errors in the computed constants, and particularly the rate constant k, may be considerably magnified with respect to the errors in the experimental data. The quantitative relationships that are presented are useful aids in the design of biochemical experiments in which the above equation is applicable.     

Evaluating the performance of a successive-approximations approach to parameter optimization in maximum-likelihood phylogeny estimation

Almost all studies that estimate phylogenies from DNA sequencedata under the maximum-likelihood (ML) criterion employ an approximateapproach. Most commonly, model parameters are estimated on someinitial phylogenetic estimate derived using a rapid method (neighbor-joiningor parsimony). Parameters are then held constant during a treesearch, and ideally, the procedure is repeated until convergenceis achieved. However, the effectiveness of this approximationhas not been formally assessed, in part because doing so requirescomputationally intensive, full-optimization analyses. Here,we report both indirect and direct evaluations of the effectivenessof successive approximations. We obtained an indirect evaluationby comparing the results of replicate runs on real data thatuse random trees to provide initial parameter estimates. Forsix real data sets taken from the literature, all replicateiterative searches converged to the same joint estimates oftopology and model parameters, suggesting that the approximationis not starting-point dependent, as long as the heuristic searchesof tree space are rigorous. We conducted a more direct assessmentusing simulations in which we compared the accuracy of phylogeniesestimated using full optimization of all model parameters oneach tree evaluated to the accuracy of trees estimated via successiveapproximations. There is no significant difference between theaccuracy of the approximation searches relative to full-optimizationsearches. Our results demonstrate that successive approximationis reliable and provide reassurance that this much faster approachis safe to use for ML estimation of topology.     

Prediction of regulation relationship between protein interactions in signaling networks

The discovery of regulation relationship of protein interactions is crucial for the mechanism research in signaling network. Bioinformatics methods can be used to accelerate the discovery of regulation relationship between protein interactions, to distinguish the activation relations from inhibition relations. In this paper, we describe a novel method to predict the regulation relations of protein interactions in the signaling network. We detected 4,417 domain pairs that were significantly enriched in the activation or inhibition dataset. Three machine learning methods, logistic regression, support vector machines(SVMs), and naïve bayes, were explored in the classifier models. The prediction power of three different models was evaluated by 5-fold cross-validation and the independent test dataset. The area under the receiver operating characteristic curve for logistic regression, SVM, and naïve bayes models was 0.946, 0.905 and 0.809, respectively. Finally, the logistic regression classifier was applied to the human proteome-wide interaction dataset, and 2,591 interactions were predicted with their regulation relations, with 2,048 in activation and 543 in inhibition. This model based on domains can be used to identify the regulation relations between protein interactions and furthermore reconstruct signaling pathways.     

Age estimation from dental eruption in infant and juvenile baboons (Papio sp.)

Two regression methods are proposed for estimating age in nonhuman primates from deciduous dental eruption data. The first method consists of step-wise multiple regression using dental eruption state (present/absent) of each tooth as independent variables. The second method uses the total number of teeth erupted as an independent variable in an exponential model. We applied both methods to a sample of 175 well nourished infant and juvenile baboons (Papio sp.), housed in an outdoor breeding corral, and ranging in age from birth to 763 days. From this sample, 129 animals were used to compute the regression formulae, and 46 animals were used for cross validation. Both models show good overall fits and high predictive accuracy with the independent cross validation sample.