A microcomputer method for designing optimal experiments for estimating enzyme kinetic parameters

A computer program, written in BASIC, for designing optimal experiments with the aim of evaluating estimates of the parameters for any enzyme kinetic model is given. This computer program can be run on any microcomputer with less than 32 Kbytes of random access memory. The program uses the termed D-optimization design criterion, which minimizes the determinant of the variance-covariance matrix. The user only supplies the rate equation, the maximum and minimum concentrations of substrates and inhibitors, the weighting pattern, and the best possible values of the parameters. The computer supplies the optimal substrate and inhibitor concentrations (one for each parameters), for estimating the parameter values, and the determinant of the variance-covariance matrix. Likewise, the microcomputer supplies the eigenvalues and eigenvectors of information and redundancy matrices, the sensitivity and the global redundancy.     

An application of multivariate ratio methods for the analysis of a longitudinal clinical trial with missing data.

This paper presents an analysis of a longitudinal multi-center clinical trial with missing data. It illustrates the application, the appropriateness, and the limitations of a straightforward ratio estimation procedure for dealing with multivariate situations in which missing data occur at random and with small probability. The parameter estimates are computed via matrix operators such as those used for the generalized least squares analysis of catetorical data. Thus, the estimates may be conveniently analyzed by asymptotic regression methods within the same computer program which computes the estimates, provided that the sample size is sufficiently computer program which computes the estimates, provided that the sample size is sufficiently large.     

Display of three-parametric data acquired by a flow cytometer

A program for producing contour maps of three-parameter flow cytometric data is described. The program is written in Pascal and can be implemented on any microcomputer with a graphics screen.     

Phylogenetic dating with confidence intervals using mean path lengths

The mean path length (MPL) method, a simple method for dating nodes in a phylogenetic tree, is presented. For small trees the age estimates and corresponding confidence intervals, calibrated with fossil data, can be calculated by hand, and for larger trees a computer program gives the results instantaneously (a Pascal program is available upon request). Necessary input data are a rooted phylogenetic tree with edge lengths (internode lengths) approximately corresponding to the number of substitutions between the nodes. Given this, the MPL method produces relative age estimates with confidence intervals for all nodes of the tree. With the age of one or several nodes of the tree being known from reference fossils, the relative age estimates induce absolute age estimates and confidence intervals of the nodes of the tree. The MPL method relies on the assumptions that substitutions occur randomly and independently in different sites in the DNA sequence and that the substitution rates are approximately constant in time, i.e., assuming a molecular clock. A method is presented for identification of the nodes in the tree at which significant deviations from the clock assumption occur, such that dating may be done using different rates in different parts of the tree. The MPL method is illustrated with the Liliales, a group of monocot flowering plants.     

The relative gonadal index: an alternative index for quantification of reproductive condition

Gonadal indices (i.e. GSI = gonadal wt/body wt X 100) commonly are used to quantify reproductive condition in fishes. These indices may be inappropriate with specimens of different sizes, however, for gonadal growth often is allometric. A new gonadal index (relative gonadal index, RGI) was developed to quantify the reproductive condition of animals independent of body size. The RGI is based on the underlying model W = alpha i X S beta i, where W is gonadal weight, S is body size (less gonadal weight if body weight is used), and alpha i and beta i are parameters to be estimated for gonadal developmental stage i. Assuming that a multiplicative lognormal error is appropriate, parameter estimates for alpha i and beta i were obtained by linear least squares regression for the log-transformed model ln(W) = beta i X ln(S) + ln(alpha i), where, in this form, beta i is the slope and ln(alpha i) is the intercept. Only if estimates of beta i do not differ significantly among ovarian developmental stages, as in our case, can a pooled estimate of beta be used to obtain the relative gonadal index, RGI = alpha i = W/S beta. Applicability of the RGI was tested using ovaries of three ecologically distinct fish species. The RGI was found to be more appropriate than the gonosomatic index for all three species.     

A calculator program for clinical application of the Bayesian method of predicting plasma drug levels

A pharmacokinetic program that allows individualization of drug dosage regimens through the Bayesian method is described. The program, which is designed for the Hewlett-Packard HP-41 CV calculator, is based upon the one-compartment open model with either instantaneous or zero-order absorption. Individualized estimation of the patient's kinetic parameters (clearance and volume of distribution) is performed by analyzing the plasma levels measured in the patient as well as considering the population data of the drug. After estimating the individual kinetic parameters by the Bayesian method, the program predicts the dosage regimen that will elicit the desired peak and trough plasma levels at steady state. For comparison purposes, the least-squares estimates for clearance and volume of distribution are calculated, and dosage prediction can also be made on the basis of the least-squares estimates. The least-squares estimates can be used to calculate population pharmacokinetic parameters according to the Standard Two-Stage method. Several examples of clinical use of the program are presented. The examples refer to patients with classic hemophilia who were treated with Factor VIII concentrates. In these patients, the Bayesian kinetic parameters of Factor VIII have been estimated through the calculator program. The Bayesian parameter estimates generated by the HP-41 have been compared with those determined by a Bayesian program (ADVISE) designed for microcomputers.     

Quantile regression models with multivariate failure time data

As an alternative to the mean regression model, the quantile regression model has been studied extensively with independent failure time data. However, due to natural or artificial clustering, it is common to encounter multivariate failure time data in biomedical research where the intracluster correlation needs to be accounted for appropriately. For right-censored correlated survival data, we investigate the quantile regression model and adapt an estimating equation approach for parameter estimation under the working independence assumption, as well as a weighted version for enhancing the efficiency. We show that the parameter estimates are consistent and asymptotically follow normal distributions. The variance estimation using asymptotic approximation involves nonparametric functional density estimation. We employ the bootstrap and perturbation resampling methods for the estimation of the variance-covariance matrix. We examine the proposed method for finite sample sizes through simulation studies, and illustrate it with data from a clinical trial on otitis media.     

Genomic growth curves of an outbred pig population

In the current post-genomic era, the genetic basis of pig growth can be understood by assessing SNP marker effects and genomic breeding values (GEBV) based on estimates of these growth curve parameters as phenotypes. Although various statistical methods, such as random regression (RR-BLUP) and Bayesian LASSO (BL), have been applied to genomic selection (GS), none of these has yet been used in a growth curve approach. In this work, we compared the accuracies of RR-BLUP and BL using empirical weight-age data from an outbred F2 (Brazilian Piau X commercial) population. The phenotypes were determined by parameter estimates using a nonlinear logistic regression model and the halothane gene was considered as a marker for evaluating the assumptions of the GS methods in relation to the genetic variation explained by each locus. BL yielded more accurate values for all of the phenotypes evaluated and was used to estimate SNP effects and GEBV vectors. The latter allowed the construction of genomic growth curves, which showed substantial genetic discrimination among animals in the final growth phase. The SNP effect estimates allowed identification of the most relevant markers for each phenotype, the positions of which were coincident with reported QTL regions for growth traits.     

Analysis of Colony-Forming Ability of Human Fibroblast Strains by Linear Regression

A linear regression approach is presented for the statistical analysis of dose-response curves obtained by measuring the colony-forming ability of human fibroblast strains. The crucial determination of the dose range in which the linear model can be assumed is achieved by a combination of statistical criteria and biological claims. As a basic quantitative parameter we investigate the slope of the regression line and, by taking reciprocals, we retransform it into the biologically established parameter D₀. Several methods for the combination of estimates are presented.     

Pedigree-free animal models: the relatedness matrix reloaded

Animal models typically require a known genetic pedigree to estimate quantitative genetic parameters. Here we test whether animal models can alternatively be based on estimates of relatedness derived entirely from molecular marker data. Our case study is the morphology of a wild bird population, for which we report estimates of the genetic variance-covariance matrices (G) of six morphological traits using three methods: the traditional animal model; a molecular marker-based approach to estimate heritability based on Ritland's pairwise regression method; and a new approach using a molecular genealogy arranged in a relatedness matrix (R) to replace the pedigree in an animal model. Using the traditional animal model, we found significant genetic variance for all six traits and positive genetic covariance among traits. The pairwise regression method did not return reliable estimates of quantitative genetic parameters in this population, with estimates of genetic variance and covariance typically being very small or negative. In contrast, we found mixed evidence for the use of the pedigree-free animal model. Similar to the pairwise regression method, the pedigree-free approach performed poorly when the full-rank R matrix based on the molecular genealogy was employed. However, performance improved substantially when we reduced the dimensionality of the R matrix in order to maximize the signal to noise ratio. Using reduced-rank R matrices generated estimates of genetic variance that were much closer to those from the traditional model. Nevertheless, this method was less reliable at estimating covariances, which were often estimated to be negative. Taken together, these results suggest that pedigree-free animal models can recover quantitative genetic information, although the signal remains relatively weak. It remains to be determined whether this problem can be overcome by the use of a more powerful battery of molecular markers and improved methods for reconstructing genealogies.     

EZ-FIT: a practical curve-fitting microcomputer program for the analysis of enzyme kinetic data on IBM-PC compatible computers

EZ-FIT, an interactive microcomputer software package, has been developed for the analysis of enzyme kinetic and equilibrium binding data. EZ-FIT was designed as a user-friendly menu-driven package that has the facility for data entry, editing, and filing. Data input permits the conversion of cpm, dpm, or optical density to molar per minute per milligram protein. Data can be fit to any of 14 model equations including Michaelis-Menten, Hill, isoenzyme, inhibition, dual substrate, agonist, antagonist, and modified integrated Michaelis-Menten. The program uses the Nelder-Mead simplex and Marquardt nonlinear regression algorithms sequentially. A report of the results includes the parameter estimates with standard errors, a Student t test to determine the accuracy of the parameter values, a Runs statistic test of the residuals, identification of outlying data, an Akaike information criterion test for goodness-of-fit, and, when the experimental variance is included, a chi 2 statistic test for goodness-of-fit. Several different graphs can be displayed: an X-Y, a Scatchard, an Eadie-Hofstee, a Lineweaver-Burk, a semilogarithmic, and a residual plot. A data analysis report and graphs are designed to evaluate the goodness-of-fit of the data to a particular model.     

Selection criterion of work matrix as a function of limiting estimates of the covariance matrix of correlated data in GEE

The modeling of generalized estimating equations used in the analysis of longitudinal data whether in continuous or discrete variables, necessarily requires the prior specification of a correlation matrix in its iterative process in order to obtain the estimates of the regression parameters. Such a matrix is called working correlation matrix and its incorrect specification produces less efficient estimates for the model parameters. Due to this fact, this study aims to propose a selection criterion of working correlation matrix based on the covariance matrix estimates of correlated responses resulting from the limiting values of the association parameter estimates. For validation of the criterion, we used simulation studies considering normal and binary correlated responses. Compared to some criteria in the literature, it was concluded that the proposed criterion resulted in a better performance when the correlation structure for exchangeable working correlation matrix was considered as true structure in the simulated samples and for large samples, the proposed criterion showed similar behavior to the other criteria, resulting in higher success rates.     

Instrumental noise estimates stabilize and quantify endothelial cell micro-impedance barrier function parameter estimates

A novel transcellular micro-impedance biosensor, referred to as the electric cell-substrate impedance sensor or ECIS, has become increasingly applied to the study and quantification of endothelial cell physiology. In principle, frequency dependent impedance measurements obtained from this sensor can be used to estimate the cell–cell and cell–matrix impedance components of endothelial cell barrier function based on simple geometric models. Few studies, however, have examined the numerical optimization of these barrier function parameters and established their error bounds. This study, therefore, illustrates the implementation of a multi-response Levenberg–Marquardt algorithm that includes instrumental noise estimates and applies it to frequency dependent porcine pulmonary artery endothelial cell impedance measurements. The stability of cell–cell, cell–matrix and membrane impedance parameter estimates based on this approach is carefully examined, and several forms of parameter instability and refinement illustrated. Including frequency dependent noise variance estimates in the numerical optimization reduced the parameter value dependence on the frequency range of measured impedances. The increased stability provided by a multi-response non-linear fit over one-dimensional algorithms indicated that both real and imaginary data should be used in the parameter optimization. Error estimates based on single fits and Monte Carlo simulations showed that the model barrier parameters were often highly correlated with each other. Independently resolving the different parameters can, therefore, present a challenge to the experimentalist and demand the use of non-linear multivariate statistical methods when comparing different sets of parameters.     

A set of programs for analysis of kinetic and equilibrium data

A program package that can be used for analysis of a wide rangeof kinetic and equilibrium data is described. The four programswere written in Turbo Pascal and run on PC, XT, AT and compatibles.The first of the programs allows the user to fit data with 16predefined and one user-defined function, using two differentnon-linear least-squares procedures. Two additional programsare used to test both the evaluation of model functions andthe least-squares fits. One of these programs uses two simpleprocedures to generate a Gaussian-distributed random variablethat is used to simulate the experimental error of measurements.The last program simulates kinetics described by differentialequations that cannot be solved analytically, using numericalintegration. This program helps the user to judge the validityof steady-state assumptions or treatment of kinetic measurementsas relaxations. Received on September 19, 1989; accepted on March 16, 1990     

Statistical analysis of nonlinear parameter estimation for Monod biodegradation kinetics using bivariate data

A nonlinear regression technique for estimating the Monod parameters describing biodegradation kinetics is presented and analyzed. Two model data sets were taken from a study of aerobic biodegradation of the polycyclic aromatic hydrocarbons (PAHs), naphthalene and 2-methylnaphthalene, as the growth-limiting substrates, where substrate and biomass concentrations were measured with time. For each PAH, the parameters estimated were: q(max), the maximum substrate utilization rate per unit biomass; K(S), the half-saturation coefficient; and Y, the stoichiometric yield coefficient. Estimating parameters when measurements have been made for two variables with different error structures requires a technique more rigorous than least squares regression. An optimization function is derived from the maximumlikelihood equation assuming an unknown, nondiagonal covariance matrix for the measured variables. Because the derivation is based on an assumption of normally distributed errors in the observations, the error structures of the regression variables were examined. Through residual analysis, the errors in the substrate concentration data were found to be distributed log-normally, demonstrating a need for log transformation of this variable. The covariance between ln C and X was found to be small but significantly nonzero at the 67% confidence level for NPH and at the 94% confidence level for 2MN. The nonlinear parameter estimation yielded unique values for q(max), K(S), and Y for naphthalene. Thus, despite the low concentrations of this sparingly soluble compound, the data contained sufficient information for parameter estimation. For 2-methylnaphthalene, the values of q(max) and K(S) could not be estimated uniquely; however, q(max)/K(S) was estimated. To assess the value of including the relatively imprecise biomass concentration data, the results from the bivariate method were compared with a univariate method using only the substrate concentration data. The results demonstrated that the bivariate data yielded a better confidence in the estimates and provided additional information about the model fit and model adequacy. The combination of the value of the bivariate data set and their nonzero covariance justifies the need for maximum likelihood estimation over the simpler nonlinear least squares regression.     

A computer program for enzyme kinetics that combines model discrimination, parameter refinement and sequential experimental design.

A method of model discrimination and parameter estimation in enzyme kinetics is proposed. The experimental design and analysis of the model are carried out simultaneously and the stopping rule for experimentation is deduced by the experimenter when the probabilities a posteriori indicate that one model is clearly superior to the rest. A FORTRAN77 program specifically developed for joint designs is given. The method is very powerful, as indicated by its usefulness in the discrimination between models. For example, it has been successfully applied to three cases of enzyme kinetics (a single-substrate Michaelian reaction with product inhibition, a single-substrate complex reaction and a two-substrate reaction). By using this method the most probable model and the estimates of the parameters can be obtained in one experimental session. The FORTRAN77 program is deposited as Supplementary Publication SUP 50134 (19 pages) at the British Library (Lending Division), Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem. J. (1986) 233, 5.     

MODELING BRAIN EVOLUTION FROM BEHAVIOR: A PERMUTATIONAL REGRESSION APPROACH

This paper has two complementary purposes: first, to present a method to perform multiple regression on distance matrices, with permutation testing appropriate for path-length matrices representing evolutionary trees, and then, to apply this method to study the joint evolution of brain, behavior and other characteristics in marsupials. To understand the computation method, consider that the dependent matrix is unfolded as a vector y; similarly, consider X to be a table containing the independent matrices, also unfolded as vectors. A multiple regression is computed to express y as a function of X. The parameters of this regression (R² and partial regression coefficients) are tested by permutations, as follows. When the dependent matrix variable y represents a simple distance or similarity matrix, permutations are performed in the same manner as the Mantel permutational test. When it is an ultrametric matrix representing a dendrogram, we use the double-permutation method (Lapointe and Legendre 1990, 1991). When it is a path-length matrix representing an additive tree (cladogram), we use the triple-permutation method (Lapointe and Legendre 1992). The independent matrix variables in X are kept fixed with respect to one another during the permutations. Selection of predictors can be accomplished by forward selection, backward elimination, or a stepwise procedure. A phylogenetic tree, derived from marsupial brain morphology data (28 species), is compared to trees depicting the evolution of diet, sociability, locomotion, and habitat in these animals, as well as their taxonomy and geographical relationships. A model is derived in which brain evolution can be predicted from taxonomy, diet, sociability and locomotion (R² = 0.75). A new tree, derived from the “predicted” data, shows a lot of similarity to the brain evolution tree. The meaning of the taxonomy, diet, sociability, and locomotion predictors are discussed and conclusions are drawn about the evolution of brain and behavior in marsupials.     

Identification of insulin receptor systems: assessing the impact of model selection and measurement error on precision of parameter estimates using Monte Carlo study

An extensive Monte Carlo study has been carried out in order to study the effect of measurement error on the precision of parameter estimates of an insulin binding system. Hypothetical radioimmunoassay experiments were generated for insulin binding to erythrocytes. The design of experiments followed strictly the protocol of real experiments. Randomly generated error was added to the synthetic data. The standard technique, a weighted non-linear regression analysis, was employed to re-estimate parameters of a model of two receptor sites and a model of negative co-operativity. As the original parameter values were known, the differences between original and estimated values was studied for (a) measurement error in the range from 0-17%, (b) random initial estimates and (c) error-free non-specific binding. In addition, analytical estimates of parameter precision were compared with the true between-experiment variation of parameter estimates. At the measurement error of 12%, a one site model is recommended to estimate the high affinity population of the two sites model. Plausible results can be expected in 90% of experiments, the between-experiment variation being approximately 30%. The model of two receptor sites gives approximately two thirds of plausible results. The high affinity population can be estimated with the between-experiment variation of 40%, the low affinity population is virtually unidentifiable with the between-experiment variation of approximately 100% and parameter estimates biased to higher values. Only half of the results obtained from the model of negative co-operativity are plausible, the variation in parameter estimates ranges from 90-150% and estimates are biased to higher values. At the level of 12% measurement error, random initial estimates do not significantly affect the estimation process, provided initial estimates are selected from a feasible range. At the same measurement error, the error-free non-specific binding does not improve the results, indicating that the mean of six replicates may be taken as a reliable estimate of non-specific binding. The analytical estimates of the coefficient of variation systematically underestimates the true between-experiments coefficient of variation, the difference has been found to be about 50%.     

PACBERG: an adaptive program for controlling the blood sugar

PACBERG, a new computer program for automatically controlling (clamping) the blood sugar is described. The plasma glucose concentration is maintained at steady basal (euglycemic) or elevated (hyperglycemic) levels during various insulin infusions. To accomplish this, a minimal mathematical model of glucose kinetics is implemented on a minicomputer. From the measured time course of plasma glucose concentration, using the model, the program estimates the insulin-dependent increase in fractional disappearance rate of glucose (X). In addition the program calculates the rate of exogenous glucose infusion (INF) which must be infused in order to maintain the desired glucose concentration. The program successfully clamps glucose at desired basal or elevated levels. Furthermore, the variable X, provided as the PACBERG studies proceed, is a real-time measure of insulin action which can be used to calculate insulin sensitivity (SI). PACBERG was written in BASIC, but can also be implemented on programmable calculators.     

Multiple Imputation Approaches for the Analysis of Dichotomized Responses in Longitudinal Studies with Missing Data

Summary Often a binary variable is generated by dichotomizing an underlying continuous variable measured at a specific time point according to a prespecified threshold value. In the event that the underlying continuous measurements are from a longitudinal study, one can use the repeated‐measures model to impute missing data on responder status as a result of subject dropout and apply the logistic regression model on the observed or otherwise imputed responder status. Standard Bayesian multiple imputation techniques ( Rubin, 1987 , in Multiple Imputation for Nonresponse in Surveys) that draw the parameters for the imputation model from the posterior distribution and construct the variance of parameter estimates for the analysis model as a combination of within‐ and between‐imputation variances are found to be conservative. The frequentist multiple imputation approach that fixes the parameters for the imputation model at the maximum likelihood estimates and construct the variance of parameter estimates for the analysis model using the results of Robins and Wang (2000, Biometrika 87, 113–124) is shown to be more efficient. We propose to apply ( Kenward and Roger, 1997 , Biometrics 53, 983–997) degrees of freedom to account for the uncertainty associated with variance–covariance parameter estimates for the repeated measures model.