Graphical analyses on binding of ligands to a two-sited system. Theoretical treatments exemplified on yeast phosphoglycerate kinase

Equilibrium studies on ATP^4? and 3-P-glycerate binding to phosphoglycerate kinase (ATP:3-phospho-d-glycerate 1-phosphotransferase, EC 2.7.2.3) have been performed. The results show that the enzyme contains two binding sites for both ligands, as was earlier suggested to explain some of the kinetic results. As long as structural information is lacking analysis of binding data can be analyzed if assumptions are made whether the sites are equivalent or nonequivalent, and whether ligation of these sites is mutually dependent or not. To understand the experimental results explicit expressions were derived for the limiting slopes and intercepts of Klotz and Scatchard plots, respectively, in terms of the intrinsic affinity constants valid under the separate assumptions made. The algebraic expressions for the limiting slopes were analyzed in order to ascertain the relationship between the form of these graphical plots, the prevailing types of binding site, and the intrinsic affinity constants. The results show that in case the equivalence of the binding sites is questionable it is necessary to be careful when the type of interaction is being determined. In certain cases it might be difficult to distinguish between positive and negative interaction. Sites that appear to be equivalent and independent could equally well be nonequivalent and dependent.     

Segregation and linkage analysis for longitudinal measurements of a quantitative trait

We present a method for using slopes and intercepts from a linear regression of a quantitative trait as outcomes in segregation and linkage analyses. We apply the method to the analysis of longitudinal systolic blood pressure (SBP) data from the Framingham Heart Study. A first-stage linear model was fit to each subject's SBP measurements to estimate both their slope over time and an intercept, the latter scaled to represent the mean SBP at the average observed age (53.7 years). The subject-specific intercepts and slopes were then analyzed using segregation and linkage analysis. We describe a method for using the standard errors of the first-stage intercepts and slopes as weights in the genetic analyses. For the intercepts, we found significant evidence of a Mendelian gene in segregation analysis and suggestive linkage results (with LOD scores >or= 1.5) for specific markers on chromosomes 1, 3, 5, 9, 10, and 17. For the slopes, however, the data did not support a Mendelian model, and thus no formal linkage analyses were conducted.     

Interaction of protein with a self-associating ligand. Deviation from a hyperbolic binding curve and the appearance of apparent co-operativity in the Scatchard plot

We derived a general formula to analyze a binding system in which a ligand self-associates, in terms of experimentally determinable quantities, i.e. r, the average number of bound ligands per protein molecule, and Lft, the total free ligand concentration, which are expressed as a ligand monomer unit. The limiting behaviors of the Scatchard plot (r/Lft vs r plot), that is, the intercepts on the r-axis and the r/Lft-axis, and the limiting slopes, are generally given. Three models that may be encountered are considered in detail. Numerical examples are also presented to illustrate how the self-association of a ligand affects the binding curves. The ligand self-association alone can cause deviation of the profile of the binding curve (r vs Lft plot) from a hyperbola, resulting in a nonlinear Scatchard plot. Therefore, analysis of the binding data without consideration of ligand self-association may lead to erroneous conclusions as to the numbers and classes of binding sites, co-operativity among the sites and binding parameter values.     

A two-stage model for multiple time series data of counts

We propose a two-stage model for time series data of counts from multiple locations. This method fits first-stage model(s) using the technique of iteratively weighted filtered least squares (IWFLS) to obtain location-specific intercepts and slopes, with possible lagged effects via polynomial distributed lag modeling. These slopes and/or intercepts are then taken to a second-stage mixed-effects meta-regression model in order to stabilize results from various locations. The representation of the models from the stages into a combined mixed-effects model, issues of inference and choices of the parameters in modeling the lag structure are discussed. We illustrate this proposed model via detailed analysis on the effect of air pollution on school absenteeism based on data from the Southern California Children's Health Study.     

An extended Scatchard analysis for competitively bound ligands and its application to cyclic adenosine-3',5'-monophosphate and cyclic 5'-amido-5'-deoxyadenosine-3', 5'-monophosphate binding to beef skeletal muscle protein.

A method for determining the dissociation constants of ligands and ligand analogs is described. It is based on competition binding studies in the presence of an isotope-labeled, or otherwise measurable, ligand and suitable analog concentrations.The steps used are determination of (1) the maximal amount of radioactive ligand that can be bound, (2) the slopes and intercepts from Scatchard plots at different analog concentrations and (3) the values for the dissociation constants of radioactive ligand and ligand analog from replots of the reciprocals of the slopes and intercepts obtained from the Scatchard plots. Application of the method to a cyclic AMP-binding protein from beef muscle is demonstrated, yielding dissociation constants of 2.10-9 M for cyclic (3H) AMP and cyclic AMP, and 3.10-5 M for cyclic 5'-amido-5'-deoxyadenosine-3', 5'-monophosphate.     

Some general aspects regarding the interpretation of binding data by means of a Scatchard plot

The question is quantitatively examined which general information can be obtained from experimental data of ligand binding to macromolecules if represented by a Scatchard plot. In particular, the curvature as well as the intercepts and slopes at both coordinate axes are analyzed assuming rather general conditions including cooperative behavior (with applications to some simplified cases of interest in practice). The results should provide a basis for attempts to elucidate the binding mechanism of a system encountered in experimental work.     

Theoretical aspects of DNA-protein interactions: co-operative and non-co-operative binding of large ligands to a one-dimensional homogeneous lattice

The interaction of proteins binding non-specifically to DNA, as well as the properties of many other interacting ligand-lattice systems important in molecular biology, requires a fundamentally different type of theoretical analysis than that provided by the classical Scatchard independent-binding-site treatment. Exact and relatively simple equations describing the binding of both non-interacting and interacting (co-operative) ligands to a homogeneous one-dimensional lattice are derived in terms of ligand site size, intrinsic binding constant and ligand-ligand co-operativity (equations (10) and (15) in the text). The mathematical approach is based on simple conditional probabilities, and reveals some largely unrecognized characteristics of such lattice binding systems. The results indicate that the binding of any non-interacting ligand covering more than one lattice residue results in non-linear (convex downward) Scatchard plots. The introduction of positive ligand-ligand co-operativity antagonizes this non-linearity, and eventually leads to plots of the opposite curvature. The maxima, limiting slopes, and intercepts of such plots can be used to estimate the required binding parameters. The method can be extended to systems involving heterogeneous ligands, and some types of heterogeneous lattices. Procedures for applying the method to a variety of interacting systems are presented, and a preliminary analysis is carried out for some selected sets of data from the literature.     

Comparison of subjects' perception of inspiratory and expiratory resistance

Six healthy male adults were studied at five levels of suprathreshold added resistance (delta R) applied thrice to either inspiration (I) or expiration (E) in a random sequence. Subjects squeezed on isometric handgrip dynamometer to express the perceived magnitude of the load. Peak mouth pressure (Pm), flow, grip (G), and delta R were analyzed to derive the exponent for Steven's power law. We observed that the slope for log G vs. log delta R was significantly greater for I loads than for E loads (P less than 0.05), but the intercepts for E loads were significantly elevated. However, the slopes and intercepts for log G vs. log Pm during the same I and E loads were not significantly different. When subjects were instructed to target I or E flow to a preset level, we observed no difference between the slopes and intercepts for log G vs. log delta R during I and E loading. These results suggest that 1) the sensory information utilized in judging the magnitude of added resistance is more likely related to the force generated by the respiratory muscles (Pm) rather than delta R per se; and 2) similar muscle receptors and neural processing systems are utilized in the estimation of added loads involving either inspiratory or expiratory muscle groups.     

Multiple imputation methods for handling incomplete longitudinal and clustered data where the target analysis is a linear mixed effects model

Multiple imputation (MI) is increasingly popular for handling multivariate missing data. Two general approaches are available in standard computer packages: MI based on the posterior distribution of incomplete variables under a multivariate (joint) model, and fully conditional specification (FCS), which imputes missing values using univariate conditional distributions for each incomplete variable given all the others, cycling iteratively through the univariate imputation models. In the context of longitudinal or clustered data, it is not clear whether these approaches result in consistent estimates of regression coefficient and variance component parameters when the analysis model of interest is a linear mixed effects model (LMM) that includes both random intercepts and slopes with either covariates or both covariates and outcome contain missing information. In the current paper, we compared the performance of seven different MI methods for handling missing values in longitudinal and clustered data in the context of fitting LMMs with both random intercepts and slopes. We study the theoretical compatibility between specific imputation models fitted under each of these approaches and the LMM, and also conduct simulation studies in both the longitudinal and clustered data settings. Simulations were motivated by analyses of the association between body mass index (BMI) and quality of life (QoL) in the Longitudinal Study of Australian Children (LSAC). Our findings showed that the relative performance of MI methods vary according to whether the incomplete covariate has fixed or random effects and whether there is missingnesss in the outcome variable. We showed that compatible imputation and analysis models resulted in consistent estimation of both regression parameters and variance components via simulation. We illustrate our findings with the analysis of LSAC data.     

Effect of the incomplete separation of bound and free ligand on binding measurements

An incomplete separation of free and acceptor-bound ligand causes underestimation of specific binding even though the determinations are corrected with blank or nonspecific binding values. If the failure of the experimental procedure causes contamination of bound ligand with free ligand, Scatchard plots are linear, though their slopes and abscissa intercepts are different from the true ones. On the other hand, if the ligand-acceptor complex is incompletely recovered, Scatchard plots are curvilinear with downward concavity. These problems can be overcome if the separable fractions of free and bound ligand are measured and suitable corrections are applied.The separation of free and receptor-bound ¹²⁵I-labeled human growth hormone by a precipitation method is taken as an example of the procedure.     

The pH dependence of the inhibition of ascorbate oxidase by anions

Double-reciprocal plots of azide inhibition, with respect to ascorbate, of ascorbate oxidase indicate mixed-type inhibition at pH values above 6. This is in contrast to the simple competitive inhibition previously observed at pH 5.6. Linear replots of the slopes and intercepts of the double-reciprocal plots yield two inhibition constants. Both constants are pH dependent. Similar inhibition patterns are obtained with fluoride and thiocyanate. These results suggest the presence of two inhibitor binding sites, one of which is competitive with respect to ascorbate and the other uncompetitive.     

Total length‐weight relationships among Thai freshwater fishes and the influence of capture location and preservation

Total length‐weight regressions on logarithmic transformed data are provided for 33 species, with 26 apparently being first estimates. All regressions are highly significant with slope values between 2.5720 and 3.4436. Slopes and intercepts of length‐weight regressions did not differ significantly with location for each of three species. Preservation in 10% formalin for 10 days followed by 30 days in 70% ethanol did not significantly influence regression slopes for the three species, length ranges and/or locations examined (ancova ). Significance among regression intercepts was equivocal with ancova analyses mostly suggestive of differences and overlap of 95% confidence limits mostly suggestive of similarities.     

Fluorescence energy transfer in one dimension: Frequency-domain fluorescence study of DNA–fluorophore complexes

The theory for the salt dependence of the free energy, entropy, and enthalpy of a polyelectrolyte in the PB (PB) model is extended to treat the nonspecific salt dependence of polyelectrolyte–ligand binding reactions. The salt dependence of the binding constant (K) is given by the difference in osmotic pressure terms between the react ants and the products. For simple 1-1 salts it is shown that this treatment is equivalent to the general preferential interaction model for the salt dependence of binding [C. Anderson and M. Record (1993) Journal of Physical Chemistry, Vol. 97, pp. 7116–7126]. The salt dependence, entropy, and enthalpy are compared for the PB model and one specific form of the preferential interaction coefficient model that uses counterion condensation/limiting law (LL) behavior. The PB and LL models are applied to three ligand–polyelectrolyte systems with the same net ligand charge: a model sphere–cylinder binding reaction, a drug–DNA binding reaction, and a protein–DNA binding reaction. For the small ligands both the PB and limiting law models give (ln K vs. In [salt]) slopes close in magnitude to the net ligand charge. However, the enthalpy/entropy breakdown of the salt dependence is quite different. In the PB model there are considerable contributions from electrostatic enthalpy and dielectric (water reorientation) entropy, compared to the predominant ion cratic (release) entropy in the limiting law model. The relative contributions of these three terms in the PB model depends on the ligand: for the protein, ion release entropy is the smallest contribution to the salt dependence of binding. The effect of three approximations made in the LL model is examined: These approximations are (1) the ligand behaves ideally, (2) the preferential interaction coefficient of the polyelectrolyte is unchanged upon ligand binding, and (3) the polyelectrolyte preferential interaction coefficient is given by the limiting law/counterion-condensation value. Analysis of the PB model shows that assumptions 2 and 3 break down at finite salt concentrations. For the small ligands the effects on the slope cancel, however, giving net slopes that are similar in the PB and LL models, but with a different entropy/enthalpy breakdown. For the protein ligand the errors from assumptions 2 and 3 in the LL model do not cancel. In addition, the ligand no longer behaves ideally due to its complex structure and charge distribution. Thus for the protein the slope is no longer related simply to the net ligand charge, and the PB model gives a much larger slope than the LL model. Additionally, in the PB model most of the salt dependence of the protein binding comes from the change in ligand activity, i.e. from nonspecific anion effects, in contrast to the small ligand case. While the absolute binding is sensitive to polyelectrolyte length, little length effect is seen on the salt dependence for the small ligands at 0.1M salt, and for lengths > 60 Å. Almost no DNA length dependenceis seen in the salt dependence of the protein binding, since this is determined primarily by the protein, not the DNA. © 1995 John Wiley & Sons, Inc.     

Towards a coherent allometric framework for individual home ranges,key population patches and geographic ranges

Spatial requirements of species have often been related to body size, usually focusing on one area variable and one taxonomic group at a time. Here, we carried out a quantitative meta‐analysis and developed a minimal model, covering different types of spatial characteristics and several species groups. In a global literature review, 46 empirical regressions on home ranges and geographic ranges were collected, covering thousands of species monitored in various countries. In addition, regional data on minimum key population patches of 167 species occurring in the Netherlands were retrieved from reports. To check consistency, a theoretical model was derived from rate and density variables based on energy equivalency. The minimum number of individuals needed to sustain a viable population was considered invariant to size. According to the equations, areas were expected to scale to mass with an exponent of 1 for the individual home range and of about ¾ for the minimum population ranges. The meta‐analysis of the empirical regressions showed that average slopes for individual home ranges were between 0.74 for cold‐blooded species and 1.05 for birds and mammals. Minimum and average species geographic range scaled to mass with exponents of 1.16–1.29 and 0.28–0.46 respectively. Allometric correlations for the minimum key patch area were weak. The intercepts indicated that carnivores require more space than equally sized herbivores, while homeotherms occupy larger areas than heterotherms. Observed slopes and intercepts were often near model estimations, but important deviations from the average level were noted as well, especially for birds. Although variability was substantial in some cases, allometric approaches can contribute considerably to understanding and protecting area requirements of species.     

Intracellular compartmentalization of adenosine triphosphate

The intracellular distribution and diffusivity of adenosine triphosphate (ATP) was studied by cryomicrodissection of individual Rana pipiens oocytes. We measured ATP concentrations in the nucleus, in animal and vegetal hemisphere cytoplasm, and in an intracellular reference phase (iRP, a microinjected gelatin "organelle") which samples diffusive ATP. Regional concentrations were not equal: nucleus much greater than animal ooplasm greater than vegetal ooplasm. ATP binding and water availability (as solvent) were determined by plotting nuclear and cytoplasmic ATP concentrations as a function of reference phase ATP concentrations (isothermal analysis). The nucleus/iRP isotherm for ATP was an equimolar line, showing that nucleoplasm resembles iRP gelatin (and consequently a simple aqueous solution) in its solvent properties. Cytoplasm/iRP isotherms were more complex, having slopes much less than unity and ordinal intercepts above the graph's origin. They demonstrate the presence in cytoplasm of mechanisms that are capable of excluding and binding ATP. These mechanisms are responsible for the inhomogeneity in ATPs intracellular distribution. In addition, exclusion and binding have different and opposing effects on ATP concentrations in the cell's "soluble space," and hence on ATP availability to enter into cellular reactions. It follows that these phenomena must be considered in attempts to model ATPs role in metabolism.     

A Random Intercepts–Functional Slopes Model for Flexible Assessment of Susceptibility in Longitudinal Designs

Summary In many biomedical investigations, a primary goal is the identification of subjects who are susceptible to a given exposure or treatment of interest. We focus on methods for addressing this question in longitudinal studies when interest focuses on relating susceptibility to a subject's baseline or mean outcome level. In this context, we propose a random intercepts–functional slopes model that relaxes the assumption of linear association between random coefficients in existing mixed models and yields an estimate of the functional form of this relationship. We propose a penalized spline formulation for the nonparametric function that represents this relationship, and implement a fully Bayesian approach to model fitting. We investigate the frequentist performance of our method via simulation, and apply the model to data on the effects of particulate matter on coronary blood flow from an animal toxicology study. The general principles introduced here apply more broadly to settings in which interest focuses on the relationship between baseline and change over time.     

Adenosine and ATP: mutually exclusive modifiers of the kinetics of ADP-induced aggregation of calf-platelets

Modulations of initial rate kinetics of ADP-induced aggregation of citrated calf platelet-rich plasma by adenosine and ATP were investigated employing a spectrophotometric platelet aggregation assay. The data were analysed according to the tenets of sequential shape-change and interaction model of aggregation. Adenosine and ATP increased the slopes and intercepts of double-reciprocal plots of ADP-aggregation kinetics. Examination of their slope and intercept effects together with their effects individually and in combination, on aggregation rates, suggested that adenosine and ATP acted at multiple, nonoverlapping, sites.     

Adjusting the phenology model of codling moth (lepidoptera: tortricidae) in Washington State apple orchards

Studies were conducted with codling moth, Cydia pomonella L., to fit cumulative curves for the occurrence of injured fruits and male moth catches in sex pheromone-baited traps as a function of accumulated degree-days after the start of moth flight. Twelve data sets were collected from seven apple, Malus domestica Bordhausen, orchards in Washington State from 2003 to 2006. Cumulative data were grouped across years for orchards either treated with sex pheromone dispensers or untreated and fit to logistic equations for both the first and second generation. No significant differences were found for the cumulative curves of moth flight or egg hatch between pheromone-treated and untreated orchards; thus, these data were combined. These new logistic models for moth flight and egg hatch were compared with a widely used distributed-delay model originally developed in Michigan. The cumulative flight curves for the logistic and distributed-delay models were statistically different (slopes) for the first but not the second generation. Cumulative egg hatch in the logistic model was significantly different from the distributed-delay model (intercepts and slopes) for both generations. Most strikingly, the timing of 50% egg hatch during the first generation was delayed 100 DD in the logistic model. The potential impact of this change in the characterization of codling moth's phenology on the effectiveness of insecticide programs targeting eggs and newly eclosed larvae was examined. Possible explanations for this significant difference between the models are discussed.     

Competitive binding assays for riboflavin and riboflavin-binding protein

A competitive binding procedure that can be used to determine either riboflavin or riboflavin-binding protein has been developed. Riboflavin-binding protein from chicken egg white binds tightly to DEAE-cellulose while free riboflavin does not. Stock [2-¹⁴C]riboflavin solutions, diluted with varying amounts of a standard unlabeled riboflavin solution or an unknown sample, are mixed with aporiboflavin-binding protein and washed through small DEAE-cellulose columns. The protein-bound riboflavin is batch eluted into scintillation vials, counted, and the unknown samples compared to a standard curve. This is a simple, rapid method for assaying riboflavin by isotope dilution. By a slight modification of the incubation conditions of this procedure, the degree of saturation and amount of riboflavin-binding protein can be determined. Data from both assays can be represented by linear plots in which slopes or intercepts correspond to unknown values. The principles presented here have been extended to the assay of biotin and avidin and should apply to other vitamins and vitamin-binding proteins.