Nonlinear Least-squares Analysis Research Articles

Evaluation of Adair binding constants from experimental data obtained with an Imai cell

One group of laboratories uses an 'even-weighted', or unweighted, nonlinear least-squares method for the analysis of experimental oxygen binding data obtained with an Imai oxygenation apparatus. Another group uses an 'end-weighted' nonlinear least-squares analysis. With end weighting each observation is assigned a statistical weight which is proportional to Y(1-Y), where Y is the fractional saturation. In this work we discuss statistical weighting functions as applied to the Imai oxygenation apparatus and then determined what are the best weighting factors for an actual series of published experiments. Based on these calculations, it is concluded that the 'best' weighting for the Imai oxygenation apparatus is a very small amount of end weighting. Furthermore, the amount of end weighting is so small that even weighting, or no weighting at all, is also appropriate.

Graphical Analysis of Reversible Radioligand Binding from Time—Activity Measurements Applied to [N-11C-Methyl]-(−)-Cocaine PET Studies in Human Subjects

A graphical method of analysis applicable to ligands that bind reversibly to receptors or enzymes requiring the simultaneous measurement of plasma and tissue radioactivities for multiple times after the injection of a radiolabeled tracer is presented. It is shown that there is a time t after which a plot of integral of t0ROI(t')dt'/ROI(t) versus integral of t0Cp(t')dt'/ROI(t) (where ROI and Cp are functions of time describing the variation of tissue radioactivity and plasma radioactivity, respectively) is linear with a slope that corresponds to the steady-state space of the ligand plus the plasma volume,.Vp. For a two-compartment model, the slope is given by lambda + Vp, where lambda is the partition coefficient and the intercept is -1/[kappa 2(1 + Vp/lambda)]. For a three-compartment model, the slope is lambda(1 + Bmax/Kd) + Vp and the intercept is -[1 + Bmax/Kd)/k2 + [koff(1 + Kd/Bmax)]-1) [1 + Vp/lambda(1 + Bmax/Kd)]-1 (where Bmax represents the concentration of ligand binding sites and Kd the equilibrium dissociation constant of the ligand-binding site complex, koff (k4) the ligand-binding site dissociation constant, and k2 is the transfer constant from tissue to plasma). This graphical method provides the ratio Bmax/Kd from the slope for comparison with in vitro measures of the same parameter. It also provides an easy, rapid method for comparison of the reproducibility of repeated measures in a single subject, for longitudinal or drug intervention protocols, or for comparing experimental results between subjects. Although the linearity of this plot holds when ROI/Cp is constant, it can be shown that, for many systems, linearity is effectively reached some time before this. This analysis has been applied to data from [N-methyl-11C]-(-)-cocaine ([11C]cocaine) studies in normal human volunteers and the results are compared to the standard nonlinear least-squares analysis. The calculated value of Bmax/Kd for the high-affinity binding site for cocaine is 0.62 +/- 0.20, in agreement with literature values.

Resolution of multicomponent fluorescence emission using frequency-dependent phase angle and modulation spectra.

We describe a new fluorescence method that allows the resolution of both the decay times and emission spectra of mixtures of fluorophores. This method is completely general and does not require any assumptions or knowledge of the decay times or emission spectra of the individual fluorophores. We use the phase angle spectra and modulation spectra of the mixture, measured over a range of suitable light modulation frequencies and emission wavelengths. These data are analyzed by nonlinear least-squares analysis to recover the emission spectra and the associated decay times. The principle of the method and the nature of the data are illustrated by using two-component mixtures with increasing spectral overlap. We then demonstrate the recovery of minor components, of structure emission spectra, and of a three-component mixture with completed overlapping emission spectra. And finally, we describe the resolution of a two-component mixture with decay times of 0.8 and 1.4 ns using modulation frequencies up to 774 MHz.

Comparative least-squares analysis of hemoglobin oxygen equilibrium curves

The oxygen-binding properties of hemoglobin have been studied at 600 microM protein concentration with organic phosphate, and analyzed by a series of different nonlinear least-squares analysis methods to determine whether reports of negligibly small values of the third overall Adair parameter, A3, are consequences of the data or a product of the data analysis. Data from other laboratories were analyzed as well. The single most important factor in creating a measurement that yields a small A3 is the use of equally weighted fitting in the Adair equation, while end-weighted fitting generally yields a larger A3. Endpoint extrapolation is ruled out as a major cause of abnormal A3 values. Monte Carlo simulations of the 600 microM results suggest that, if a small A3 were present, end weighting is at least as sensitive to a small A3 as equal weighting. We conclude that equally weighted fitting of the tetrameric Adair equation is unable to resolve the upper asymptote of the oxygen-binding data, resulting in an unusually small value for A3.

Off-resonance rotating frame spin-lattice NMR relaxation studies of phosphorus metabolite rotational diffusion in bovine lens homogenates

The rotational diffusion behavior of phosphorus metabolites present in calf lens cortical and nuclear homogenates was investigated by the NMR technique of 31P off-resonance rotating frame spin-lattice relaxation as a means of assessing the occurrence and extent of phosphorus metabolite-lens protein interactions. 31P NMR spectra of calf lens homogenates were obtained at 10 and 18 degrees C (below and above the cold cataract phase transition temperature, respectively) at 7.05 T. Effective rotational correlation times (tau 0,eff) for the major phosphorus metabolites present in cortical and nuclear bovine calf lens homogenates were derived from nonlinear least-squares analysis of R vs omega e (spectral intensity ratio vs precessional frequency about the effective field) data with the assumption of isotropic reorientational motion. Intramolecular dipole-dipole (1H-31P, 31P-31P), chemical shift anisotropy (CSA), and solvent (water) translational intermolecular dipole-dipole (1H-31P) relaxation contributions were assumed in the analyses. In those cases where the limiting value of the spectral intensity ratio failed to reach unity at large offset frequency, a modified formalism incorporating chemical exchange mediated saturation transfer between two sites was used. Values of tau 0,eff for phosphorus metabolites present in the cortex varied from a low of ca. 2 ns [L-alpha-glycero-phosphocholine (GPC)] to a high of 12 ns (alpha-ATP) at 10 degrees C, whereas at 18 degrees C the range was from ca. 1 to 9 ns. For the nucleus the tau 0,eff values ranged from ca. 3 ns (GPC) to 41 ns (Pi) at 10 degrees C; at 18 degrees C the corresponding values ranged from 4 to 39 ns. For PME (phosphomonoester; in lens the predominant metabolite is L-alpha-glycerol phosphate) at 18 degrees C evidence was obtained for binding and subsequent exchange with solid like protein domains. The diversity in tau 0,eff values for lenticular phosphorus metabolites is suggestive of differential binding to more slowly tumbling macromolecular species, most likely lens crystallin proteins. Corresponding measurement of tau 0,eff values for the mobile protein fraction present in calf lens cortical and nuclear homogenates at 10 and 18 degrees C, by 13C off-resonance rotating frame spin-lattice relaxation, provided average macromolecular correlation times that were assumed to represent the bound metabolite state. A fast-exchange model (on the T1 time scale), between free and bound forms, was employed in the analysis of the metabolite R vs omega e curves to yield the

Kinetic Model of 2-Deoxyglucose Metabolism Using Brain Slices

A six-compartment, nine-parameter kinetic model of 2-deoxyglucose (2DG) metabolism, which includes bidirectional tissue transport, phosphorylation, two-step dephosphorylation, phosphoisomerization, and conjugation to UDP and macromolecules, has been derived. Data for analysis were obtained from 540- and 1,000-microns-thick hippocampal and hypothalamic brain slices, which were incubated in buffer containing [14C]2DG, frozen, extracted with perchlorate, and separated on anion-exchange columns. Solutions of the equations of the model were fit to the data by means of nonlinear least-squares analysis. These studies suggest that dephosphorylation is adequately described by a single reaction so that the model reduces to eight parameters. The in vitro rate constants for transport, phosphorylation, and dephosphorylation are very similar to prior in vivo results. The phosphoisomerization rate constant is similar to dephosphorylation, so glycosylated macromolecules slowly accumulate and gradually assume larger relative importance as other compounds disappear more rapidly. Rate constants for 540-microns slices from hypothalamus and hippocampus are similar, while 1,000-microns slices have smaller tissue transport constants and larger phosphorylation constants. The rate equation for glucose utilization of this model is relatively insensitive to uncertainties regarding the rate constants. Including later metabolic components in kinetic models improves the calculations of glucose utilization with long isotope exposures.

Rotational energy transfer in CH3F: The ΔJ=n, ΔK=0 processes

We report the measurement of the rates o ΔJ=n, ΔK=0(‖n‖≤10) processes for CH3F–CH3F collisions at 300 K. The data are derived from a time-resolved millimeter/submillimeter-infrared double resonance investigation of both the 12CH3F and the 13CH3F isotopic species. The rates were obtained via a nonlinear least-squares analysis of the data using a numerical simulation of rotational energy transfer in methyl fluoride. These rates are shown to be quantifiable in terms of the scaling law of infinite order sudden collision theory and the statistical power gap law. As a result, the numerous ΔJ=n, ΔK=0(‖n‖>1) rates can be understood in terms of only two parameters, independent of isotopic species. Using these results and the results of our earlier studies of K-changing processes, we discuss how rotational energy transfer in the CH3F system in general can be described in terms of a small number of collisional processes and parameters.

COSIFIT: An interactive program for simultaneous multioscillator cosinor analysis of time-series data

A BASIC program for the Macintosh family of computers has been developed that provides an iterative nonlinear least-squares analysis of biological rhythm data using Marquardt's modification of the Gauss-Newton algorithm. A cosinor model is used, and multiple cosine functions can be fit simultaneously to any equi- or unequispaced time-series data. The program computes optimal frequency, mesor, amplitude, and phase, with standard errors of measurement and both parametric and nonparametric estimates of goodness-of-fit. Multiple time-series can be analyzed simultaneously, allowing one to test for shared characteristics between series. Statistical differences between parameter values of select curves are ascertained by ANOVA. The program is interactive and designed to be user-friendly for the scientist or clinician.

Intensity-based domain refinement of oriented but unpositioned molecular replacement models

A program is described that performs least-squares group refinement of oriented molecular replacement models whose positions in the unit cell are unknown. The program (INTREF) is designed to produce improved models for use in a translation function by optimizing the orientations and relative translations of the model domains. The molecular contents of the asymmetric unit are refined as a small number of rigid bodies whose origins relative to each other may be unknown. More than one molecule in the asymmetric unit can be accommodated. The refinement seeks to minimize the residual error between the observed and calculated intensities that have been modified to produce the equivalent of a radial weighting in Patterson space. Calculated intensities include contributions from all symmetry-related molecules, enabling meaningful refinement in high-symmetry space groups. Derivatives of the intensities with respect to the rigid-body parameters are evaluated numerically using fast Fourier transforms and the shifts are obtained by non-linear least-squares analysis. Results with test cases show that the program is capable of adjusting the orientations and relative translations of protein domains to give models that more closely resemble the known structures. Consequently, the resulting models produce more accurate and more interpretable results in translation functions. The importance of including all crystallographically related molecules and of downweighting the contribution of the longer-radius region of the Patterson function is demonstrated.

Application of mathematical model to experimental chemotherapy of fatal murine pneumonia

Two beta-lactam antibiotics, cefazolin and cefmenoxime, were administered in an experimental model of murine pneumonia caused by Klebsiella pneumoniae in a way which enabled us to approximate the serum antibiotic concentration time course in humans. Bacterial counts during the experiments were subjected to nonlinear least-squares analyses by using a mathematical model that explained the bacterial killing by the antibiotic concentration time course and other factors associated with antimicrobial potency and bacterial growth. Cefazolin gave a killing curve that changed synchronously with the drug levels in serum; in contrast, cefmenoxime gave a curve that was prolonged as compared with the change in the drug levels in serum. Multiple correlation coefficients were about 0.9, and the model worked well for bacterial count data. Parameters relating to antimicrobial potency of the drugs, bacterial growth rate, and drug distribution into the tissue were estimated numerically.

Specific binding of .BETA.-endorphin to the isolated renal basolateral membranes in vitro.

Binding of human beta-endorphin (beta-EP) to rat renal basolateral membranes was characterized using [125I]Tyr27-beta-EP ([125I]beta-EP) as a primary ligand. Ten millimolar of ethylenediaminetetra acetic acid (EDTA) completely inhibited the degradation of [125I]beta-EP in the incubation mixture at 4 degrees C, thus making it possible to quantitatively examine the [125I]beta-EP binding. The specific binding of [125I]beta-EP to the basolateral membranes was reversible and saturable, and a nonlinear least-squares regression analysis of a saturation isotherm revealed two different classes of specific binding sites. One class had an apparent dissociation constant (Kd) of 0.68 nM and a lower number of binding sites (33 fmol/mg protein), whereas the other class had a lower affinity (apparent Kd of 210 nM) and a higher number of binding sites (7.3 pmol/mg protein). Inhibition of the [125I]beta-EP binding by naloxone (10 microM) was approximately only 20%, and that by D-Ala2-D-Leu5-enkephalin (10 microM) was null, suggesting the major role of a non-opioid binding component in specific [125I]beta-EP binding to basolateral membranes. Moreover, a 50% inhibition by 10 microM of dynorphin(1-13) suggests that a certain region of the primary structure of beta-EP, excluding at least the NH2-terminal enkephalin sequence, is of particular importance for the [125I]beta-EP binding. These lines of evidence suggest the existence of two different classes of specific binding sites for beta-EP on the renal basolateral membranes, and the high-and low-affinity bindings may be attributed to opioid and non-opioid receptors, respectively, as judged by known characteristics of opioid and non-opioid receptors in other peripheral tissues.

Comparison of the Effects of Ions and GTP on Substance P Binding to Membrane‐Bound and Solubilized Specific Sites

Mg2+ increased but Na+ and GTP decrease [3H]substance P (SP) binding to rat cerebral cortical membranes and to 10 mM 3-[(3-cholamidopropyl)dimethylammonio]-1-propane sulfonate (CHAPS)-solubilized membrane fraction. To determine the binding parameters that are modified by the cations and GTP, inhibition experiments of [3H]SP binding by unlabeled SP were performed in both of the preparations. Nonlinear least-squares regression analysis of data in the membrane fraction indicated that optimal fitting of the inhibition curves in the presence of 10 mM MgCl2 was attained with a two-site model, corresponding to a "high-affinity (H)" and a "low-affinity (L)" state. By omitting MgCl2, or by addition of NaCl and GTP, the [3H]SP specific binding was decreased, the H state disappeared, and the L state and a new "super-low affinity (SL)" state observed. The SP/[3H]SP inhibition curves in the cerebral cortical membranes by in vivo treatment with pertussis toxin (islet-activating protein) were similar to that in the presence of GTP in control membranes. The effects of MgCl2, NaCl, and GTP were greater in the CHAPS-solubilized fraction than in the membrane fraction. In contrast to the membrane fraction, the inhibition curves of [3H]SP binding by unlabeled SP in the presence of MgCl2 in the CHAPS-solubilized fraction were best fitted to a one-site model. The KD value was relatively close to that of the low-affinity state in the membrane fraction. Even with the addition of NaCl or GTP, or by reducing MgCl2 concentration to 1 mM, although the inhibition curves consistently fit the one-site model, the KD values changed only slightly.

Accumulation of weak base in gastric mucosa provides evidence for an acidic storage compartment.

Uptake and release of acridine orange (AO), a fluorescent weak base that accumulates in acidic spaces, were studied in perfused frog gastric mucosa. Tissue was mounted between two flow-through chambers and loaded with AO on the mucosal side. AO washout and acid secretion rate were monitored simultaneously by a flow-through fluorescence detector and a pH-stat, respectively. Data were displayed on a computer screen, stored, and analyzed. AO, in concentrations as high as 0.02 mM, does not affect the acid secretion rate. Nonlinear least-squares analysis of AO washout curves resolved two exponential components: a faster component associated mainly with AO washout from the chamber and a slower component reflecting primarily AO washout from the tissue. The slower exponential declines more slowly at higher concentrations and/or longer duration of AO loading, whereas the faster exponential is unaffected. AO washout is unaffected by the level of the steady-state acid secretion rate. Nitrite inhibits the acid secretion rate but does not affect the AO washout. When nitrite is removed, acid secretion rate and fluorescence (AO concentration in the mucosal medium) increase simultaneously and transiently. The net amount of AO released from the tissue is proportional to the net amount of acid released. Stimulation by secretagogue in basally secreting tissue causes synchronous transient increases in acid secretion rate and fluorescence. We conclude that accumulation of AO provides evidence for the existence of an intracellular storage pool of free protons within the transporting epithelium.

Information theory and the analysis of ligand-binding data.

The phenomenological principles of information theory are used in the analysis of ligand-binding phenomena in biological macromolecules. Information maps are constructed to visualize regions of ligand chemical potential with maximum amount of information and to devise suitable experimental strategies therefrom. Extensive simulation studies and analysis of experimental data also point out the properties of information used as a weighting procedure in nonlinear least-squares analyses.

In vitro kinetics of reduction of cytochrome c554 isolated from the reaction center of the green phototrophic bacterium, Chloroflexus aurantiacus

The photochemical reaction center in the green bacterium Chlaroflexus aurantiacus is similar to that found in purple phototrophic bacteria and interacts with a multiheme membrane-bound cytochrome. We have examined the kinetics of reduction of the pure solubilized reaction center cytochrome by laser flash photolysis of solutions containing lumiflavin or FMN. Reduction by lumiflavin semiquinone followed single exponential kinetics and the observed rate constant ( k obs) was linearly dependent on protein concentration ( k = 1.8 × 10 7 m −1 s −1 heme −1). This result suggests either that the four hemes have similar reduction rate constants which cannot be resolved or that there are large differences in rate constant and only the most reactive heme (or hemes) was observed under these conditions. To determine the relative reactivities of the four hemes, we varied the extent of heme reduction at a single total protein concentration. As the hemes were progressively reduced by steady-state illumination prior to laser flash photolysis, k obs for the reaction with fully reduced lumiflavin decreased nonlinearly. Second-order rate constants for the four hemes were assigned by nonlinear least-squares analysis of k obs vs oxidized heme concentration data. The second-order rate constants obtained in this way for the highest and lowest potential hemes differed by a factor of about 20, which is larger than expected for c-type cytochromes based on redox potential alone (a factor of about 3 would be expected). This is interpreted as being due to differences in steric accessibility. Relative to the highest potential heme, which is as reactive as a typical c-type cytochrome, we estimated a steric effect of approximately twofold for heme 2, and steric effects of approximately fivefold for hemes 3 and 4. Using fully reduced FMN as reductant of oxidized cytochrome, ionic strength effects indicate a minus-minus interaction, with approximately a −2 charge near the site of reduction of the highest potential heme.

Cellulose pyrolysis kinetics in a simulated solar environment

The instrumentation and methods required to study rapid, radiative decomposition of highly reactive biopolymers are described and used to determine the kinetic parameters for the pyrolysis of Avicel PH105 microcrystalline cellulose. Experiments were carried out at heating rates greater than 2 K/s in a specially fabricated thermogravimetric analyzer system which employed a 30-kW e /2-kW th downward-facing beam, arc-image furnace as a heat source. The thermogravimetric data were interpreted assuming a simple, single-step reaction mechanism and using a nonlinear least-squares kinetic analysis

Pyrolysis of Methyl Nitrite and 1,3,5-Trioxane Mixtures in Shock Waves: Kinetic Modeling of the H + CH2O Reaction Rate

Abstract Mixtures of methyl nitrite and 1,3,5-trioxane have been pyrolyzed in shock waves at temperatures between 1180 and 1690 K. The rates of CO production measured by CO laser resonance absorption were kinetically modeled to obtain the rate constant for reaction (9): H + CH2O → CHO + H2. The result from modeling 34 sets of data gave rise to k 9 = 10l3.46exp( − 1879/T) cm3/mole-sec over the entire temperature range studied. A nonlinear least-squares analysis of these high temperature data and all published low temperature (T < 1000 K) values led to k 9 = 108.10 T 1.62exp (−1089/T)cm3/mole-sec, covering 260 ≤ T ≤ 1700 K.

Kinetic Studies on the Inhibition Mechanism of Diamine Oxidase from Porcine Kidney by Aminoguanidine

The mechanism of inhibition of diamine oxidase [DAO, histaminase, amine: oxygen oxidoreductase (deaminating) (copper-containing), EC 1.4.3.6] by aminoguanidine was studied kinetically. The DAO reaction was carried out with putrescine as a substrate in the presence of o-aminobenzaldehyde and followed by the continuous increase in A430. An enzymatic reaction product, 4-aminobutyraldehyde, was spontaneously cyclized to delta 1-pyrroline, which coupled with o-aminobenzaldehyde to give rise to a yellow quinazolinium chromophore [Holmstedt et al. (1961) Biochim. Biophys. Acta 48, 182-186]. First, to measure the initial velocities as accurately as possible, the kinetic relationship between the oxidase reaction rate and the color development rate was formulated. The reaction proceeded linearly in the absence of aminoguanidine, but slowed down time-dependently in its presence. When the color development velocities in the presence of the inhibitor were plotted against time, the plot was nonlinear and there seemed to be at least two phases of the inhibition. Nonlinear least-squares analysis of the data dA430/dt showed that the simplest model that fitted best was a model composed of two species of enzyme-inhibitor complexes: E + I k+1 in equilibrium k-1 EI k+2 in equilibrium k-2 E'I where E is the enzyme, I is aminoguanidine (an inhibitor), and the ks are the rate constants.(ABSTRACT TRUNCATED AT 250 WORDS)

Fatigue properties of acrylic bone cement: S-N, P-N, and P-S-N data.

The determination of the fatigue properties of a material is a fundamental criteria in the engineering design process. The fatigue properties are governed by a number of factors, one of which is the inherent scatter in the data. In order to take into account this scatter in the results, the concept of the probability of failure (P) is introduced and interconnected with the well known stress (S) versus number of cycles to failure (N) data. This report determines the S-N curve, P-N curve, and P-S-N contour for the three leading acrylic bone cements: Surgical Simplex P, Zimmer LVC, and Zimmer Regular. Tensile specimens were fabricated according to ASTM D638 specifications and tested in uniaxial, zero-tension fatigue. The resulting stress versus number of cycles to failure data was subjected to a nonlinear least-squares analysis to determine the mathematical expression of the fatigue curve. Statistical analysis showed excellent fit of the data to the predicted curve for estimation of the endurance limit of each cement. The results indicated that Zimmer LVC had the highest endurance limit while the limit of Simplex-P and Zimmer Regular were significantly lower. No significant difference was noted in the endurance limit between Simplex-P and Zimmer Regular. The probability of failure at each stress level was determined with respect to fatigue distribution functions. Using the normal distribution function and previous S-N data, the P-S-N contour was generated for each cement. The P-S-N contour fully describes the fatigue characteristics of the material.

Characterization of microscopic laser beams by 2-D scanning of fluorescence emission

We describe a fast, accurate, and reproducible method for calibrating microscopic laser beams. Fundamental mode (TEM(00)) output from an argon-ion laser is focused on a thin (<0.5-microm) aqueous fluorescein solution located at the sample plane of a fluorescence microscope. The fluorescent image of the focused beam is scanned in two dimensions at the emission diaphragm of the microscope by a computer-controlled X-Y scanning mirror. Plots of fluorescence intensity vs position demonstrate that the beam profile is Gaussian in two dimensions. Both the 1/e(2) beam radius and the coordinates of peak intensity are obtained by nonlinear least-squares analysis. An adaptation of this method is used to characterize microscopic elliptical Gaussian beams. We present experimental and theoretical evidence, the latter based on the optical transfer function of an objective lens, that Gaussian beams used in fluorescence photobleaching recovery experiments are not significantly distorted by diffraction or defocusing effects.