Gaussian quadrature as a numerical integration method for estimating area under the curve.

This paper presents a numerical integration method for estimating the area under the curve (AUC) over the infinite time interval. This method is based on the Gauss-Laguerre quadrature and produces AUC estimates over the infinite time interval without extrapolation in a usual sense. By contrast, in traditional schemes, piecewise interpolation is used to obtain the area up to the final sampling point, and the remaining portion is extrapolated using nonlinear regression. In this case, there is no theoretical consistency between the quadrature and extrapolation. The inconsistency may cause certain problems. For example, the optimal sampling criterion for the former is not necessarily optimal for the latter. Such inconsistency does not arise in the method of this work. The sampling points are placed near the zeros of Laguerre polynomials so as to directly estimate the AUC over the infinite time interval. The sampling design requires no particular prior information. This is also advantageous over the previous strategy, which worked by minimizing the variance of estimated AUC under the assumptions of particular pharmacokinetic and variance functions. The original Gaussian quadrature is believed to be inappropriate for numerical integration of data because of several restrictions. In this paper, it is shown that, using a simple strategy for managing errors due to these restrictions, the method produces an estimate of AUC with practically sufficient precision. The efficacy of this method is finally shown by numerical simulations in which the bias and variance of its estimate were compared with those of the previous methods such as the trapezoidal, log-trapezoidal, Lagrange, and parabolas-through-the-origin methods.

A comparison of methods for estimating individual pharmacokinetic parameters.

Characteristics of the methods for estimating individual pharmacokinetic parameters are compared both theoretically and numerically. The methods examined represent the range of most of modern methods and include the ordinary least squares, iteratively reweighted least squares, extended least squares, generalized least squares, maximum quasi-likelihood and its extended scheme, and minimum relative entropy methods. When the function representing the mean itself is used as a variance function, which may be then related to a Poisson distribution, the iteratively reweighted least squares estimator and maximum quasi-likelihood estimator are both identical to that of the minimum relative entropy method. These methods work by minimizing a kind of relative entropy between observed data and corresponding theoretical values. Furthermore, these methods guarantee agreement between the sum of the observed values and the estimate of the sum. This relation does not hold in general for the other estimators. The sum can, in a sense, be viewed as an approximation of the area under the curve. In addition, it is shown by numerical study that these methods are robust against the misspecification of the variance model and work as effectively as such sophisticated methods as the extended least squares, generalized least squares, and maximum extended quasi-likelihood methods. These sophisticated methods require complicated numerical optimization techniques and should be used only in cases where the estimation of the variance function is demanded. In the other cases, the method of minimum relative entropy or its equivalent is sufficient or even preferable for estimating individual pharmacokinetic parameters.

Extended quasi-likelihood is useful for analyzing intra-individual variability in pharmacokinetic regression models.

The method of maximum extended quasi-likelihood (MEQL) can be viewed as an estimation method in the framework of generalized linear models. The method was applied to a pharmacokinetic problem in which the pharmacokinetic model was a nonlinear function of its parameters. The behavior of the method toward the estimation of a variance function was numerically compared with those of the generalized least squares (GLS) and extended least squares methods. In general, the MEQL and GLS methods were equally better. However, the MEQL estimator often showed smaller mean squared errors for the scaling parameter than the other two estimators. Such a generally comparable but partially distinct property of the MEQL method, as compared with the GLS method, is useful to pharmacokinetic analysis.

Pharmacokinetic parameter estimations by minimum relative entropy method.

For estimating pharmacokinetic parameters, we introduce the minimum relative entropy (MRE) method and compare its performance with least squares methods. There are several variants of least squares, such as ordinary least squares (OLS), weighted least squares, and iteratively reweighted least squares. In addition to these traditional methods, even extended least squares (ELS), a relatively new approach to nonlinear regression analysis, can be regarded as a variant of least squares. These methods are different from each other in their manner of handling weights. It has been recognized that least squares methods with an inadequate weighting scheme may cause misleading results (the "choice of weights" problem). Although least squares with uniform weights, i.e., OLS, is rarely used in pharmacokinetic analysis, it offers the principle of least squares. The objective function of OLS can be regarded as a distance between observed and theoretical pharmacokinetic values on the Euclidean space RN, where N is the number of observations. Thus OLS produces its estimates by minimizing the Euclidean distance. On the other hand, MRE works by minimizing the relative entropy which expresses discrepancy between two probability densities. Because pharmacokinetic functions are not density function in general, we use a particular form of the relative entropy whose domain is extended to the space of all positive functions. MRE never assumes any distribution of errors involved in observations. Thus, it can be a possible solution to the choice of weights problem. Moreover, since the mathematical form of the relative entropy, i.e., an expectation of the log-ratio of two probability density functions, is different from that of a usual Euclidean distance, the behavior of MRE may be different from those of least squares methods. To clarify the behavior of MRE, we have compared the performance of MRE with those of ELS and OLS by carrying out an intensive simulation study, where four pharmaco-kinetic models (mono- or biexponential, Bateman, Michaelis-Menten) and several variance models for distribution of observation errors are employed. The relative precision of each method was investigated by examining the absolute deviation of each individual parameter estimate from the known value. OLS is the best method and MRE is not a good one when the actual observation error magnitude conforms to the assumption of OLS, that is, error variance is constant, but OLS always behaves poorly with the other variance models. On the other hand, MRE performs better than ELS and OLS when the variance of observation is proportional to its mean. In contrast, ELS is superior to MRE and OLS when the standard deviation of observation is proportional to its mean. In either case the difference between MRE and ELS is relatively small. Generally, the performance of MRE is comparable to that of ELS. Thus MRE provides as reliable a method as ELS for estimating pharmacokinetic parameters.

A molecular dynamics study of branched alpha-cyclodextrin.

A branched alpha-cyclodextrin is a derivative of an alpha-cyclodextrin with a branch consisting of an extra glucose unit. Its water solubility is considerably higher than that of the unbranched one. We have studied the high solubility of the molecule in aqueous solution by molecular dynamics simulations. Trajectories of the molecule at 293 K were calculated using GROMOS programs in three different environments, i.e., in vacuo, in the crystalline state, and in aqueous solution. A simulation in vacuo was carried out to explore stable conformations of the molecule in the isolated system. The quality of the simulations were examined by comparing the X-ray and the simulated crystal structures. The results of the simulations show three remarkable structural features of the molecule: self-inclusion with its branched portion, twist-boat conformation of a glucose ring, and wobbling of its macrocycle. Among these, the last feature is closely related to the water solubility of the molecule. The solubility of cyclodextrin appears to be mainly governed by its intramolecular interglucose hydrogen bonds, which inhibit hydration by solvent water molecules. The results of our simulations indicate that the capability to form hydrogen bonds in branched alpha-cyclodextrin decreased as the macrocycle of the molecule lost its regular circular shape. Such wobbling of the macrocycle was observed on a relatively short time scale (several picoseconds). An extra glucose unit introduced to alpha-cyclodextrin may cause the improved water solubility of the molecule through the greater wobbling motion of its macrocycle.

Clinical significance of benzoate-metabolizing capacity in patients with chronic liver disease: pharmacokinetic analysis.

Benzoate-metabolizing capacity was studied in control subjects and in liver disease patients after intra-venous loading of 15 mg benzoate per kg of body weight. In the 7 control subjects, the mean level (+/- SEM) of Cmax for serum benzoate was 104.1 +/- 6.8 micrograms/ml, AUC was 2.57 +/- 0.32 mg.min/ml, MRT was 21.5 +/- 1.5 min and T1/2 was 15.5 +/- 1.3 min. For serum hippurate, on the other hand, Tmax was 27.9 +/- 6.0 min, Cmax was 33.4 +/- 2.1 micrograms/ml, AUC was 1.96 +/- 0.13 mg.min/ml, MRT was 39.6 +/- 2.9 min and T1/2 was 30.7 +/- 2.4 min. In 12 patients with chronic hepatitis, Cmax, AUC, MRT and T1/2 for benzoate and Tmax, MRT and T1/2 for hippurate remained at control levels, but Cmax and AUC for hippurate were slightly decreased compared to controls. However, in 18 patients with liver cirrhosis, Cmax and AUC for benzoate were in the control range but MRT and T1/2 were significantly delayed (p less than 0.01 for both). Moreover, the MRT value was increased in proportion to the severity of liver disease (p less than 0.01). AUC for hippurate was not changed to any extent, and Tmax, MRT and T1/2 were slightly delayed, while Cmax was significantly reduced. AUC, MRT and T1/2 for benzoate and Tmax, MRT and T1/2 for hippurate showed significant correlation with serum albumin levels, prothrombin time and indocyanine green clearance rate. These results suggest that benzoate-metabolizing capacity, especially as indicated by the MRT value for serum benzoate, appears to be a better index than the indocyanine green clearance rate for determining hepatic functional reserve in chronic liver disease.

Conformation of 2'GMP bound to a mutant ribonuclease T1 (Y45W) determined by X-ray diffraction and NMR methods.

The crystal structure of a mutant ribonuclease T1 (Y45W) complexed with a specific inhibitor, 2'GMP, has been determined by X-ray diffraction and refined at 1.9 A resolution to a conventional R-factor of 0.164. The mode of recognition of the guanine base by the enzyme is similar to that found for the wild-type ribonuclease T1 complexed with 2'GMP. The binding of the guanine base is clearly enhanced by maximum overlapping of the indole ring of Trp45 and the base. The glycosyl torsion angle of the inhibitor is in the syn conformation and the sugar exhibits a C3'-endo type pucker, which differs from that observed in the crystal of the complex between the wild-type ribonuclease T1 and 2'GMP. Analysis of 500-MHZ NMR spectra has also indicated that the 2'GMP molecule as bound to the mutant enzyme in solution exhibits a C3'-endo type pucker, similar to that bound to the wild-type enzyme in solution [Inagaki, Shimada, & Miyazawa (1985) Biochemistry 24, 1013-1020].

Hydrophobic effects on protein/nucleic acid interaction: enhancement of substrate binding by mutating tyrosine 45 to tryptophan in ribonuclease T1.

Hydrophobic effects on binding of ribonuclease T1 to guanine bases of several ribonucleotides have been proved by mutating a hydrophobic residue at the recognition site and by measuring the effect on binding. Mutation of a hydrophobic surface residue to a more hydrophobic residue (Tyr45----Trp) enhances the binding to ribonucleotides, including mononucleotide inhibitor and product, and a synthetic substrate-analog trinucleotide as well as the binding to dinucleotide substrates and RNA. Enhancements on binding to non-substrate ribonucleotides by the mutation have been observed with free energy changes ranging from -2.2 to -3.9 kJ/mol. These changes are in good agreement with that of substrate binding, -2.3 kJ/mol, which is calculated from Michaelis constants obtained from kinetic studies. It is shown, by comparing the observed and calculated changes in binding free energy with differences in the observed transfer free energy changes of the amino acid side chains from organic solvents to water, that the enhancement observed on guanine binding comes from the difference in the hydrophobic effects of the side chains of tyrosine and tryptophan. Furthermore, a linear relationship between nucleolytic activities and hydrophobicity of the residues (Ala, Phe, Tyr, Trp) at position 45 is observed. The mutation could not change substantially the base specificity of RNase T1, which exhibits a prime requirement for guanine bases of substrates.

Adaptability of restrained molecular dynamics for tertiary structure prediction: application to Crotalus atrox venom phospholipase A2.

In order to assess the adaptability and/or applicability of the restrained molecular dynamics (RMD) simulation for building a possible tertiary structure of a protein from the X-ray crystal structure of a family reference protein, the tertiary structure prediction of Crotalus atrox venom phospholipase A2 (PLA2) was attempted based on the X-ray crystal structure of bovine pancreatic PLA2. For the formation of secondary and tertiary structures from the fully extended starting structure, the RMD simulation with interatomic distance restraints and torsion angle restraints, which were derived from homologous amino acid sequence regions in the reference protein, was carried out until the molecular system was fully equilibrated. The predicted tertiary structure of C. atrox venom PLA2 was compared with its X-ray crystal structure, and furthermore the utility of this method was discussed by reference to the similar tertiary structure prediction of beta-trypsin from the X-ray crystal structure of an elastase.

An alternative two stage method via the EM-algorithm for the estimation of population pharmacokinetic parameters.

There has been a considerable increase in popularity of the NONMEM method as a technique for estimating population pharmacokinetic parameters. The authors present another approach to population pharmacokinetic analysis, the alternative two stage method (ATS). ATS uses the EM-algorithm for maximizing the likelihood of variance components. The performance of ATS was compared with the NONMEM method on a microcomputer. Simulation studies showed that the precision and accuracy of estimates obtained with ATS were comparable to the NONMEM method, however, the computation time, dependences on initial estimates and convergence properties were somewhat different. ATS could be a valuable alternative to the NONMEM method for estimating population pharmacokinetic parameters in some cases.

Refined X-ray structure of the low pH form of ribonuclease T1-2'-guanylic acid complex at 1.9 A resolution.

The three-dimensional X-ray structure of the RNase T1[EC 3.1.27.3]-2'GMP complex crystallized at low pH value (4.0) was determined, and refined to 1.9 A resolution to give a final R value of 0.203. The refined model includes 781 protein atoms, 24 inhibitor atoms, and 43 solvent molecules. The imidazole rings of His27 and His40 interact with the carboxyl side chains of Glu82 and Glu58, respectively, whereas that of His92 is in contact with the main chain carbonyl oxygen of Ala75. In the complex, the ribose ring of the 2'GMP molecule adopts a C2'-endo puckering, and the exocyclic conformation is gauche(-)-gauche(+). The glycosyl torsion angle is in the syn range with an intramolecular hydrogen bond between N3 and O5', and the 2'-phosphate orientation is trans-gauche(-). The guanine base of the inhibitor is tightly bound to the base recognition site with five hydrogen bonds (N1--Glu46O epsilon 2, N2---Asn98O,O6---Asn44N, and N7 ---Asn43N delta 2/Asn43N) and is sandwiched between the phenolic ring portions of Tyr42 and Tyr45 by stacking interactions. The 2'-phosphate group interacts with Arg77N eta 2, Glu58O episilon 2, and Tyr 38O eta but not with any of the histidine residues. Arg77N eta 2 also interacts with Tyr38O eta. There is no interaction between the ribose moiety of the inhibitor and the enzyme.