Mathematics of quantitative kinetic PCR and the application of standard curves

Fluorescent monitoring of DNA amplification is the basis of real-time PCR, from which target DNA concentration can be determined from the fractional cycle at which a threshold amount of amplicon DNA is produced. Absolute quantification can be achieved using a standard curve constructed by amplifying known amounts of target DNA. In this study, the mathematics of quantitative PCR are examined in detail, from which several fundamental aspects of the threshold method and the application of standard curves are illustrated. The construction of five replicate standard curves for two pairs of nested primers was used to examine the reproducibility and degree of quantitative variation using SYBER® Green I fluorescence. Based upon this analysis the application of a single, well- constructed standard curve could provide an estimated precision of ±6–21%, depending on the number of cycles required to reach threshold. A simplified method for absolute quantification is also proposed, in which quantitative scale is determined by DNA mass at threshold.     

Identification of some cleavage sites of insulin by insulin proteinase.

To overcome the limit of the intensity of ions carrying sequence information in structural determinations of peptides by fast-atom-bombardment m.s., we have developed a method that consists in taking spectra of the peptide acid hydrolysates at different hydrolysis times. Peaks correspond to the oligomers arising from the peptide partial hydrolysis. The sequence can then be identified from the structurally overlapping fragments.     

Semisynthesis and properties of some insulin analogs

The trypsin-catalyzed coupling of bovine (Boc)₂-desoctapeptide (B23-B30)-insulin with synthetic octapeptides, H-Gly-X₂-X₃-X₄-Thr-Pro-Lys(Boc)-Thr-OH (X₂ = Phe or Ala, X₃ = Phe or Ala, X₄ = Tyr or Ala), followed by deprotection and purification produced the [Ala^B24, Thr^B30]-, [Ala^B25, Thr^B30]-, and [Ala^B26, Thr^B30]-analogs of bovine insulin in yields of 32, 35, and 32%, respectively. The biological activity of these analogs decreased in the order, normal insulin ([Thr^B30]-bovine insulin) = Ala^B26-insulin > Ala^B25-insulin > Ala^B24-insulin, as assayed for receptor binding and some other biological effects, in contrast with the corresponding Leu-analogs of human insulin, in which the activity decreased in the order, normal insulin > Leu^B24-insulin > Leu^B25-insulin. The affinity to insulin antibodies greatly diminished in both Ala^B24-insulin and Leu^B24-insulin but not in the B25-substituted analogs. The CD spectra of the Leu- and the Ala-analogs were compared with those of normal insulins to show that no apparent correlation seems to exist between the decrease in biological activity and the conformational changes observed in solution. The effects of organic solvents on the peptide-bond equilibrium and on the stability of trypsin are also discussed.     

An alternative to allosteric interactions as causes of sigmoid dose versus response curves application to glucose-induced secretion of insulin

It is shown that a signoid dose vs. response curve will occur in any biological system in which the stimulant compound must diffuse significant distances to reach responding cells, even when the cells themselves respond according to Michaelis-Menten kinetics. Isolated pancreatic islets releasing insulin in response to glucose stimulation have been used as a specific example. Since diffusion and/or other physical processes can produce global effects which could account for the sigmoidal nature of a dose vs. response curve, the existence of complex molecular mechanisms of hormone-receptor interaction can not be inferred solely from the character of a dose vs. response relation.     

Influence of carbamoylation on some analytical properties of basic polypeptides

The effect of carbamoylation on the assay or identification of histones and polylysine was investigated. Incubation with sodium cyanate decreased the positive charge on these polypeptides as judged by changes in the binding of methyl orange or the electrophoretic mobility. Histones in chromatin appeared less accessible to carbamoylation than isolated histones. Carbamoylation of proteins under conditions in which there was little or no effect on the Lowry procedure could affect their assay by methods utilizing metachromasia with Coomassie Blue G. The Bradford assay has low sensitivity for Hl histone and polylysine but this can be increased by preincubation with sodium cyanate. More extensive carbamoylation of polylysine caused decreased sensitivity which was the only response seen with core nucleosomal histones and bovine serum albumin when preincubated with sodium cyanate. It was concluded that the sensitivity for Hl histone and polylysine in assays dependent on metachromasia with Coomassie Blue G may be changed by factors which decrease the positive charge on these polypeptides.     

Evaluation of species-area functions using Sonoran Desert plant data: not all species-area curves are power functions

Ecologists have been studying the relationship between species richness and area for about a century. As area increases, more species are typically observed. Many mathematical functions have been proposed to describe the pattern of increase. Numerous researchers have assumed that the relationship is a power function despite the fact that there are many possible alternatives. There has been limited work in evaluating which species-area functions are most appropriate for field data. This study examines which of a variety of functions best describe how Sonoran Desert plant species richness of remnant habitat patches in the Phoenix metropolitan area vary with sampled area and the area of entire patches. No single species-area function was adequate for describing all empirical datasets. Sample curves of woody species were most frequently best described by the sigmoid logistic, Hill, and Lomolino functions, whereas herbaceous datasets were best fit by the sigmoid logistic or convex rational functions. A curve depicting the relationship between patch-level woody species richness and patch area was best fit by the convex exponential function. The power function provided the best fit for only one case. This study demonstrates the utility of testing alternative functions for statistical fit rather than assuming that any particular equation adequately describes the species-area relationship.     

An analytical approach to the problem of inverse optimization with additive objective functions: an application to human prehension

We consider the problem of what is being optimized in human actions with respect to various aspects of human movements and different motor tasks. From the mathematical point of view this problem consists of finding an unknown objective function given the values at which it reaches its minimum. This problem is called the inverse optimization problem. Until now the main approach to this problems has been the cut-and-try method, which consists of introducing an objective function and checking how it reflects the experimental data. Using this approach, different objective functions have been proposed for the same motor action. In the current paper we focus on inverse optimization problems with additive objective functions and linear constraints. Such problems are typical in human movement science. The problem of muscle (or finger) force sharing is an example. For such problems we obtain sufficient conditions for uniqueness and propose a method for determining the objective functions. To illustrate our method we analyze the problem of force sharing among the fingers in a grasping task. We estimate the objective function from the experimental data and show that it can predict the force-sharing pattern for a vast range of external forces and torques applied to the grasped object. The resulting objective function is quadratic with essentially non-zero linear terms.     

The use of multivariable standard curves in the radioimmunoassay of testosterone and 5alpha-dihydrotestosterone

Multivariable calibration curves have been used to enable testosterone and 5alpha-dihydrotestosterone to be assayed directly in plasma extracts without further pre-purification of the sample. Two antisera were used, both with relatively high, but different affinities for the substances measured, and with relatively low affinity towards all other substances tested. The antisera were obtained from rabbits immunized against testosterone-3-BSA and 5alpha-dihydrotestosterone-3-BSA. The technique was of adequate precision, accuracy and specificity. The last was examined by comparison of values obtained by the present method and those obtained following pre-purification by thin layer chromatography.     

A three-parameter Langmuir-type model for fitting standard curves of sandwich enzyme immunoassays with special attention to the alpha-fetoprotein assay

In a simplified approach to the reaction kinetics of enzyme-linked immunoassays, a Langmuir-type equation y = [ax/(b + x)] + c was derived. This model proved to be superior to logit-log and semilog models in the curve-fitting of standard curves. An assay for alpha-fetoprotein developed in our laboratory with a sensitivity of 2 g/liter and a between-day coefficient of variation of about 15% was used for comparison of the different models. The proposed model has the advantage that it is based on the law of mass action and that blank determinations can be included in the calculation of the curves without giving a disproportionate bias.     

Differential and analytical subfractionation of rat liver components internalizing insulin and prolactin

Receptor-mediated endocytosis of 125I-insulin and 125I-prolactin into liver parenchymal cells has been studied by quantitative subcellular fractionation. Differential centrifugation yielded three particulate fractions, N (nuclear), ML (large granule), and P (microsomes), and a final supernatant (S). Quantitative differences in the extent and rates of accumulation of 125I-insulin and 125I-prolactin into the fractions were observed. The acidotropic agent chloroquine and the microtubule disrupting agent colchicine were administered separately to rats. The agents increased significantly the T 1/2 of hormone clearance from the liver and augmented the accumulation of both ligands in the low-speed ML fraction. However, differences in the rates of accumulation of insulin and prolactin into all cell fractions were still maintained. Analytical centrifugation of each of the particulate fractions was carried out in order to determine if different endocytic components were specific to insulin or prolactin internalization. This was not the case. An "early" endosomal component of density 1.11 was identified in microsomes. A "late" endosome of density 1.10 was identified in the large granule (ML) fraction. Both endosomal components appeared to accumulate insulin and prolactin but at different rates. Marker enzyme analysis identified the presumed plasma membrane component in microsomes (density approximately 1.155). This component showed a significant difference in the rate of loss of 125I-insulin (T 1/2 approximately 4.1 min) as compared to that of 125I-prolactin (T 1/2 approximately 12.7 min). A further difference in the handling of the ligands was observed in early endosomes.(ABSTRACT TRUNCATED AT 250 WORDS)     

Influence of reagent formulation on mRNA quantification by RT-PCR using imported external standard curves

Use of an imported external standard curve is common in real-time quantitative RT-PCR. Two practical strategies for long-term experiments include importing a grand mean standard curve to all accumulated runs or using daily imported standard curves, fixing the slope at the beginning of the experiment and calibrating successive runs with curves generated from this imported slope, adding a single standard that registers the variation in the y-intercept. This study determines the influence that a change in reagent lots has on these two calibration approaches when determining mRNA copy numbers of the ornithine decarboxylase and porphobilinogen deaminase genes. Two sets of determinations were run with the use of lot A and lot B. A marked decrease in the crossing points (Cp) in the standards for both genes at all concentration levels was observed with the change in lots. A grand mean standard curve was generated for each gene and each set and comparisons between the sets were performed. Statistically significant differences were found with respect to the y-intercept but not the slope, suggesting that the change of reagent lot affected the detection sensitivity but not the efficiency of the reaction. The excellent correlation coefficients obtained for these curves for each gene were not achieved when overall data from both sets were combined to generate an overall grand mean standard curve. We conclude that when faced with a change of RT-PCR reagent lot that will affect the detection sensitivity of the method, samples should be calculated with either the daily imported standard curves or with the respective grand mean standard curve for each lot.     

An alternative to allosteric interactions as causes of sigmoid dose versus response curves. Application to glucose-induced secretion of insulin.

It is shown that a sigmoid dose vs. response curve will occur in any biological system in which the stimulant compound must diffuse significant distances to reach responding cells, even when the cells themselves respond according to Michaelis-Menten kinetics. Isolated pancreatic islets releasing insulin in response to glucose stimulation have been used as a specific example. Since diffusion and/or other physical processes can produce global effects which could account for the sigmoidal nature of a dose vs. response curve, the existence of complex molecular mechanisms of hormone-receptor interaction can not be inferred solely from the character of a dose vs. response relation.     

Observed and hypothetical leaf rust progress curves of some genotypes of wheat

Summary The hypothetical leaf rust progress curves of 15 genotypes of wheat were generated by integrating the components of slow rusting resistance determined in the glasshouse. The area under the disease progress curve (AUDPC) from the hypothetical and observed leaf rust progress curves were compared. It was found that the hypothetical AUDPC values of all the genotypes studied were smaller than their respective observed AUDPC values. Possible causes for the discrepancies in the observed and hypothetical values are discussed.