Normalization in the fitting of data by iterative methods. Application to tracer kinetics and enzyme kinetics

1. The normalization of biochemical data to weight them appropriately for parameter estimation is considered, with reference particularly to data from tracer kinetics and enzyme kinetics. If the data are in replicate, it is recommended that the sum of squared deviations for each experimental variable at each time or concentration point is divided by the local variance at that point. 2. If there is only one observation for each variable at each sampling point, normalization may still be required if the observations cover more than one order of magnitude, but there is no absolute criterion for judging the effect of the weighting that is produced. The goodness of fit that is produced by minimizing the weighted sum of squares of deviations must be judged subjectively. It is suggested that the goodness of fit may be regarded as satisfactory if the data points are distributed uniformly on either side of the fitted curve. A chi-square test may be used to decide whether the distribution is abnormal. The proportion of the residual variance associated with points on one or other side of the fitted curve may also be taken into account, because this gives an indication of the sensitivity of the residual variance to movement of the curve away from particular data points. These criteria for judging the effect of weighting are only valid if the model equation may reasonably be expected to apply to all the data points. 3. On this basis, normalizing by dividing the deviation for each data point by the experimental observation or by the equivalent value calculated by the model equation may both be shown to produce a consistent bias for numerically small observations, the former biasing the curve towards the smallest observations, the latter tending to produce a curve that is above the numerically smaller data points. It was found that dividing each deviation by the mean of observed and calculated variable appropriate to it produces a weighting that is fairly free from bias as judged by the criteria mentioned above. This normalization factor was tested on published data from both tracer kinetics and enzyme kinetics.     

Effects of solvent and solute drag on transmembrane diffusion

The present study compares and quantitates both solvent drag and solute drag forces in a system with both heteropore and homopore membranes. It is shown that tracer solute permeability can be increased if solution flow or driver solute flux is in the direction of tracer diffusion. Either force can decrease tracer permeability if the force can decrease tracer permeability if the force is opposite to the direction of tracer diffusion. The two forces can be additive or one force may reduce the effect of the other force. In the particular system quantitated, solute drag is shown to be some 300 times more effective than solvent drag on a mole-to-mole basis. The use of a number of solute pairs on other homopore and heteropore membranes confirms the finding that the two drag forces can be analyzed or manipulated in a variety of systems.     

A theorem on amplitudes of thermal atomic fluctuations in large molecules assuming specific conformations calculated by normal mode analysis

An exact theorem is proved and its implication is discussed. The theorem states that, if a large molecule, typically biological macromolecules such as proteins, undergoes small-amplitude conformational fluctuations around its native conformation in such a way that within the range of conformational fluctuations at thermal equilibrium the conformational energy surface can be approximated by a multidimensional parabola, then the mass-weighted mean-square displacement of constituent atoms is given by the sum of the contributions from each normal mode of conformational vibration, which in turn is proportional to the inverse of the square of its frequency. This theorem provides a firm theoretical basis for the fact hitherto empirically recognized in the conformational dynamics of, for instance, native proteins that very-low-frequency normal modes make dominant contributions to the conformational fluctuations at thermal equilibrium. Discussion is given on the implication of this theorem, especially on the importance of the concept of the low-frequency normal modes, even in the case where the basic assumption of the harmonicity of the energy surface does not hold.     

The flux-summation theorem and the 'evolution of dominance'

The flux-summation theorem (FST) is a central principle of metabolic control analysis. It describes how the control of flux through any metabolic pathway of arbitrary complexity is distributed among the component reaction steps. Two issues concerning the FST are discussed in this paper. First, it has been suggested that the theorem could, in principle, be inapplicable under certain conditions, i.e. the sum of the control coefficients of all the enzymes supporting a pathway could exceed unity. Such conditions have not been found in any species so far studied, so in practice the FST is always applicable. I argue that applicability of the FST is a precondition for phenotypic robustness and therefore for survival. Second, the FST provides a basis for explaining dominance that renders Fisher's ‘modifier genes’ hypothesis otiose. Some recent misunderstandings of metabolic control analysis have led to the claim that this explanation is flawed and therefore that Fisher's hypothesis can and should be reinstated. Here, these suggestions are refuted.     

Perspective: Here's to Fisher,additive genetic variance,and the fundamental theorem of natural selection

Fisher's fundamental theorem of natural selection, that the rate of change of fitness is given by the additive genetic variance of fitness, has generated much discussion since its appearance in 1930. Fisher tried to capture in the formula the change in population fitness attributable to changes of allele frequencies, when all else is not included. Lessard's formulation comes closest to Fisher's intention, as well as this can be judged. Additional terms can be added to account for other changes. The "theorem" as stated by Fisher is not exact, and therefore not a theorem, but it does encapsulate a great deal of evolutionary meaning in a simple statement. I also discuss the effectiveness of reproductive-value weighting and the theorem in integrated form. Finally, an optimum principle, analogous to least action and Hamilton's principle in physics, is discussed.     

Comparison of rat dopamine D2 receptor occupancy for a series of antipsychotic drugs measured using radiolabeled or nonlabeled raclopride tracer

Preclinical brain receptor occupancy measures have heretofore been conducted by quantifying the brain distribution of a radiolabeled tracer ligand using either scintillation spectroscopy or tomographic imaging. For smaller animals like rodents, the majority of studies employ tissue dissection and scintillation spectroscopy. These measurements can also be accomplished using liquid chromatography coupled to mass spectral detection to measure the brain distribution of tracer molecules, obviating the need for radioligands. In order to validate mass spectroscopy-based receptor occupancy methods, we examined dopamine D2 receptor dose-occupancy curves for a number of antipsychotic drugs in parallel experiments using either mass spectroscopy or radioligand-based approaches. Oral dose-occupancy curves were generated for 8 antipsychotic compounds in parallel experiments using either radiolabeled or unlabeled raclopride tracer. When curves generated by these two methods were compared and ED(50) values determined, remarkably similar data were obtained. Occupancy ED(50) values were (mg/kg): chlorpromazine, 5.1 and 2.7; clozapine, 41 and 40; haloperidol, 0.2 and 0.3; olanzapine, 2.1 and 2.2; risperidone, 0.1 and 0.4; spiperone, 0.5 and 0.4; thioridazine 9.2 and 9.5; and ziprasidone 1.4 and 2.1 (unlabeled and radiolabeled raclopride tracer, respectively). The observation that in vivo application of both techniques led to comparable data adds to the validation state of the mass spectroscopy-based approach to receptor occupancy assays.     

Use of LC/MS to assess brain tracer distribution in preclinical, in vivo receptor occupancy studies: dopamine D2, serotonin 2A and NK-1 receptors as examples

High performance liquid chromatography combined with either single quad or triple quad mass spectral detectors (LC/MS) was used to measure the brain distribution of receptor occupancy tracers targeting dopamine D2, serotonin 5-HT2A and neurokinin NK-1 receptors using the ligands raclopride, MDL-100907 and GR205171, respectively. All three non-radiolabeled tracer molecules were easily detectable in discrete rat brain areas after intravenous doses of 3, 3 and 30 microg/kg, respectively. These levels showed a differential brain distribution caused by differences in receptor density, as demonstrated by the observation that pretreatment with compounds that occupy these receptors reduced this differential distribution in a dose-dependent manner. Intravenous, subcutaneous and oral dose-occupancy curves were generated for haloperidol at the dopamine D2 receptor as were oral curves for the antipsychotic drugs olanzapine and clozapine. In vivo dose-occupancy curves were also generated for orally administered clozapine, olanzapine and haloperidol at the cortical 5-HT2A binding site. In vivo occupancy at the striatal neurokinin NK-1 binding site by various doses of orally administered MK-869 was also measured. Our results demonstrate the utility of LC/MS to quantify tracer distribution in preclinical brain receptor occupancy studies.     

The Movement of Tracers along the Petiole of Nymphoides peltatum: I. A. PRELIMINARY STUDY WITH 137Cs

It is shown that the general postulate of a mass flow with lateralescape of tracer accounts fairly satisfactorily for the well-establishedexponential fall-off in tracer activity down the axis. The shapeof the curves relating the logarithm of the activity to distanceis concave upwards, and it is suggested on the assumption ofa mass-flow model that this is evidence either of an acceleratingvelocity down the petiole, or of a falling rate of lateral leakageout of the sieve tubes. It has also been shown that no satisfactoryestimate of the linear velocity of movement can be got fromthe usual type of tracer study without further information ofa kind not usually provided, and no estimate has proved possiblein the present case.     

Flux ratio theorems for nonstationary membrane transport with temporary capture of tracer

It has been shown recently that the ratio of unidirectional tracer fluxes, passing in opposite directions through a membrane which has transport properties varying arbitrarily with the distance from a boundary, is independent of time from the very first appearance of the two outfluxes from the membrane. This surprising proposition has been proved for boundary conditions defining standard unidirectional fluxes, and then generalized to classes of time-dependent boundary conditions. The operational meaning of all the resulting theorems is that when any of them appear to be refuted experimentally, the presence of more than one parallel transport pathway (that is, of membrane heterogeneity transverse to the direction of transport) can be inferred and analyzed. Recent experimental data have been interpreted accordingly. However, the proofs of the theorems given so far have not taken into account the possibility of temporary capture of tracer at sites fixed in the membrane (including also entrances to microscopic culs-de-sac). The possible presence of such a process, which would not affect fluxes in the steady state, left a fundamental gap in the aforementioned inferences. It is shown here that all the theorems previously proved for the flux ratio under unsteady conditions remain valid when temporary capture of tracer is admitted, no matter how the rate of capture, and the probability distribution of residence times of tracer at capture sites, may depend on the distance from a membrane boundary. The validity of the aforementioned inferences from observed time-dependence of the flux ratio is thereby extended to a much wider class of membrane transport processes.     

A Multi-Tracer Push-Pull Diagnostic Test for In Situ Air Sparging Systems

This document describes the development and initial application of a multi-tracer push-pull test designed to provide near real-time point-specific measures of contaminant volatilization and aerobic biodegrada-tion rates during in situ air sparging (IAS) operation. Measured biodegradation and volatilization rates are specific to the tracers used, so the results provide relative measures useful for identifying spatial differences in treatment performance and changes in performance with changes in system operation and design. The diagnostic test involves injecting a solution containing multiple tracer compounds into the target treatment zone through a monitoring well, piezometer, or drive point. The injected solution is initially deoxygenated and can contain: (a) a nondegradable, non-volatile conservative tracer, (b) one or more nondegradable, volatile chemicals, (c) an aerobically biodegradable, nonvolatile compound, and (d) a visible dye. After some predetermined hold time, an excess quantity of groundwater is extracted from the same injection point and the change in the concentrations of the tracer compounds is measured. Volatilization and oxygen utilization rates are then estimated from mass balances on the tracer components. The development of this diagnostic tool was conducted in a controlled physical model study and then initial field tests were conducted at the U.S. Navy Hydrocarbon National Test Site (HNTS) in Port Hueneme, California. Spatial variations in oxygenation and volatilization rates were observed, with oxygenation rates varying from 0 to 51 mg-O₂/L-water/d, and tracer volatilization rates varying from 0 to 47%/d. Acetate and sulfur hexafluoride (SF₆) were used as tracers in the initial testing, and it was discovered that these are not ideal choices due to the potential for anaerobic acetate biodegradation and SF₆ partitioning into trapped gas in the aquifer.     

Violations of the ceiling principle: exact conditions and statistical evidence.

The National Research Council recommended the use of the ceiling principle in forensic applications of DNA testing on the grounds that the ceiling principle was believed to be "conservative," giving estimates greater than or equal to the actual genotype frequencies in the appropriate reference population. We show here that the ceiling principle can fail to be conservative in a population with two subpopulations and two loci, each with two alleles at Hardy-Weinberg equilibrium, if there is some linkage disequilibrium between loci. We also show that the ceiling principle can fail in a population with two subpopulations and a single locus with two alleles if Hardy-Weinberg equilibrium does not hold. We give explicit analytical formulas to describe when the ceiling principle fails. By showing that the ceiling principle is not always mathematically reliable, this analysis gives users of the ceiling principle the responsibility of demonstrating that it is conservative for the particular data with which it is used. Our reanalysis of VNTR data bases of the FBI provides compelling evidence of two-locus associations within three major ethnic groups (Caucasian, black, and Hispanic) in the United States, even though the loci tested are located on different chromosomes. Before the ceiling principle is implemented, more research should be done to determine whether it may be violated in practice.     

Interpretation of relative potencies, relative efficacies and apparent affinity constants of agonist drugs estimated from concentration-response curves

Differences in the relative potencies of agonists have been used successfully in the past to classify receptors. Such use of agonists can be justified on the basis of ideas and equations developed using the occupancy model of drug action. However the occupancy model makes no allowance for possible complications which may arise when the drug-receptor complex interacts with a transducer-effector system. For some receptor-effector systems use of an equilibrium ternary complex model may be better than use of the occupancy model but the former still does not take into account the possible effect of guanosine-5'-triphosphate on the system. A steady-state version of the ternary complex model has therefore been analysed to explore possible interpretations of relative potencies, relative efficacies and apparent affinity constants estimated from concentration-response curves. It is concluded that for agonists which act on receptors which function through G-proteins these pharmacological parameters may depend on the concentration of the relevant G-protein in the cell membranes and on the intracellular concentrations of guanosine-5'-triphosphate and guanosine-5'-diphosphate. If these concentrations vary appreciably between tissues then the parameters are also likely to vary, even for a single receptor-transducer system. It follows that the use of such agonist parameters to classify receptors or receptor-transducer systems is not likely to be totally dependable. It is also possible that agonists which interact with only one receptor-transducer system may show selectivity between tissues with different concentrations of G-proteins and of guanine nucleotides.     

Basic studies on metabolic steady state. Incompletely mixed systems

This paper is the second of a pair dealing with some mathematical properties of metabolic steady state. An investigator wishing to compute the rate of appearance and/or disappearance of a metabolite in steady state within an intact biological system will usually appeal to a method involving radioactive tracers. It is shown that while the investigator’s choice of the mode of tracer administration (constant infusion or single injection) is largely arbitrary, the mathematical interpretation of the results may depend upon the presence or absence of gradients in certain of the variables of the system. The latter will be the case if the system is sampled at a point within the distribution space of the metabolite which is not a source point but is otherwise arbitrary. In order to deduce a formula which gives the required rate, he must have knowledge of the gradient of concentration of the traced substance, and sometimes of the gradient of specific activity.     

Dominance, Pleiotropy and Metabolic Structure

It is a common observation that most mutants have similar dominance relations for all the characters they are known to affect. As a model of pleiotropic effects we investigate a branched pathway where the two outputs represent two characters whose variation is affected by changes in any of the genetically specified enzymes in the system. We consider the effects on the phenotype (fluxes or intermediate metabolites) of substitutions at one locus represented by enzyme activities of the two homozygotes (mutant and wild type) and that of the heterozygote. Dominance indices for the characters pleiotropically connected by the metabolic system are calculated. We show that if enzymes behave 'linearly,' (first order), that is if saturation and feedback inhibition or other nonlinearities are absent, all fluxes and pools have identical dominance relations. The presence of such nonlinearity, however, leads to differences in dominance between different characters and we define the conditions where such differences can be important.     

Theoretical Analysis of Net Tracer Flux Due to Volume Circulation in a Membrane with Pores of Different Sizes : Relation to solute drag model

When osmotic pressure across an artificial membrane, produced by a permeable electrically neutral solute on one side of it, is balanced by an external pressure difference so that there is no net volume flow across the membrane, it has been found that there will be a net flux of a second electrically neutral tracer solute, present at equal concentrations on either side of the membrane, in the direction that the "osmotic" solute diffuses. This has been ascribed to solute-solute interaction or drag between the tracer and the osmotic solutes. An alternative model, presented here, considers the membrane to have pores of different sizes. Under general assumptions, this "heteroporous" model will account for both the direction of net tracer flux and the observed linear dependence of unidirectional tracer fluxes on the concentration of the osmotic solute. The expressions for the fluxes of solutes and solvent are mathematically identical under the two models. An inequality is derived which must be valid if the solute interaction model and/or the heteroporous model can account for the data. If the inequality does not hold, then the heteroporous model alone cannot explain the data. It was found that the inequality holds for most published observations except when dextran is the osmotic solute.     

A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis. II. The effects of non-equilibrated ligand binding

Cells generally chemotax along a direction in which their receptor occupancy gradient--whether spatial or temporal--is maximum. Occupancy differentials are, however, often so small as to be masked by thermal noise; i.e., by fluctuations inherent in the stochastic nature of ligand binding. Such fluctuations therefore impose a fundamental limit on the sensitivity of a cell's ability to detect a chemoattractant gradient. In order to pursue the implications of this limit, fluctuation theories have been developed. The theories assume that the signal is some function of the receptor occupancy gradient, allow an estimate of the standard deviation about the mean signal, and permit an evaluation of, among other things, the extent to which a receptor defect can impair an effective response. Previous theories have assumed an equilibrated ligand-receptor interaction. In this paper we introduce a generalized definition of a signal caused by a receptor occupancy gradient that allows us to develop a non-equilibrium theory of thermal noise. We show that previous formulations are a special case of the current development. More specifically, we find the following. Swimming cells subject to Brownian tumbling must generally average their signals over a very long time period to achieve a signal-to-noise ratio less than or equal to 1. Spatial gradient detection is possible with ligand-receptor equilibrium constants less than 10(3)M-1, but since such ligands are rare, theory predicts that tumbling cells will generally not detect gradients by measuring spatial occupancy differentials. These conclusions hold irrespective of whether chemical equilibrium is achieved. For crawling cells not subject to Brownian tumbling, a range of affinities exists in which spatial or temporal gradient detection is possible. In general a spatial mechanism is more efficient for low affinity ligands (dissociation times less than 0.3 s), whereas a temporal mechanism is more efficient for higher K. In this case the detection of gradients in slowly dissociating ligand will be facilitated if signal processing begins prior to chemical equilibration. An important new parameter is indicated by the theory. The definitions of a temporal gradient signal is based on estimating and comparing average occupancy over two time intervals displaced by a time t1. The theory predicts an optimal t1, of order milliseconds, that leads to the shortest minimum averaging time. For t1 values at and longer than the optimum, and for all averaging times exceeding some minimum, the cell will detect a temporal signal.(ABSTRACT TRUNCATED AT 400 WORDS)     

Polypeptide hormone receptors: characteristics and applications.

Major developments in the area of polypeptide hormone receptors have been reviewed. Receptors are high affinity, high specificity binding sites which appear to be located largely, if not entirely, on the plasma membrane of cells. Receptors are proteins intimately associated with and influenced by lipids. Receptor sites and degrading sites appear to be readily distinguishable entities. The binding of hormone to receptor is distinct and has been dissociated from subsequent steps leading to hormonal response. There is no direct relationship between receptor occupancy and the magnitude of target response to hormone. So called 'spare' receptors can be viewed thermodynamically as enhancing target tissue sensitivity to hormone. The binding of hormone to receptor appears to be a point at which regulation of tissue sensitivity can be influenced either through altering the affinity for hormone or the number of receptors. One factor apparently involved in the regulation of receptor levels is the hormone itself. Receptors have been used to develop assay procedures which have significantly complemented the bioassay and radioimmunoassay. Finally, the measurement of receptor levels in disease has provided new insights into pathophysiology.     

Single channel gating events in tracer flux experiments I. Theory

Tracer ion flux measurements are a commonly used method for studying ion transport through membranes of cellular systems, where the rate of ion flow is determined by gating processes which control the opening and closing of transmembrane channels. Due to recent advances in the theoretical analysis of tracer flux from or into closed membrane structures (CMS), the mechanism of gating reactions can, in principle, be derived from flux data. A physically well founded analysis is presented for the dependence of the total tracer ion content of a collection of CMS on the gating processes. For functionally uncoupled gating units a mean single channel flux contribution [equation, see text] can be defined, where k is the intrinsic single channel flux coefficient, t the time over which flux is measured, and p(tau,t) is the probability that a given channel was open for a total period tau during t. This quantity reflects the mean time course of the tracer content due to flux through a single channel. Expressions for are derived that explicitly take into account a distribution in the lifetime of open channels. On the basis of the results, kinetic and thermodynamic parameters of multiphasic gating reactions can be determined from the time course of the overall tracer content in a colleciion of CMS.     

Cell lineage of zebrafish blastomeres. I. Cleavage pattern and cytoplasmic bridges between cells

Patterns of cleavage and cytoplasmic connections between blastomeres in the embryo of the zebrafish, Brachydanio rerio have been described. The cell division pattern is often very regular; in many embryos a blastomere's lineage may be ascertained from its position in the cluster through the 64-cell stage. At the 5th cleavage, however, significant variability in pattern is observed, and alternative patterns of the 5th cleavage are described. The early cleavages are partial, incompletely separating blastomeres from the giant yolk cell. The tracer fluorescein-dextran (FD) was injected into blastomeres to learn the extent of the cytoplasmic bridging. It was observed that until the 10th cleavage, blastomeres located along the blastoderm margin maintain cytoplasmic bridges to the yolk cell. Beginning with the 5th cleavage, FD injected into a nonmarginal blastomere either remains confined to the injected cell, or if the injection was early in the cell cycle, the tracer spreads to the cell's sibling, through a bridge persisting from the previous cleavage. On the other hand, injected Lucifer yellow spreads, presumably via gap junctions, widely among blastomeres in a pattern unrelated to lineage.