Two New Regulatory Properties Arising from the Transient Phase Kinetics of Monocyclic Enzyme Cascades

Taking as starting point a previous contribution about the kinetics of the transient phase and steady-state of monocyclic enzyme cascades, this paper suggest the definition and use of new regulatory modification properties involving the time elapses from the onset of the reaction to the attainment of the steady-state for a monocyclic enzyme cascade. A minimal set of simplifying assumptions allowing to derive analytical expressions for these properties has been used. From these general expressions we derive, as particular cases, other simpler expressions by using additional assumptions which have, therefore, a smaller range of validity. A discussion of the relationships between the kinetic parameters and concentrations needed to the additional assumption is observed is carried out. The goodness of our analysis has been tested by using numerical integration of the set of differential equation describing the kinetic behaviour of the cascade. The results obtained for a type of cascade are extrapolable to other different schemes of monocyclic enzyme cascades. Finally, a kinetic data analysis and an experimental design are suggested     

New theoretical expressions for the five adsorption type isotherms classified by BET based on statistical physics treatment

New theoretical expressions to model the five adsorption isotherm types have been established. Using the grand canonical ensemble in statistical physics, we give an analytical expression to each of five physical adsorption isotherm types classified by Brunauer, Emett, and Teller, often called BET isotherms. The establishment of these expressions is based on statistical physics and theoretical considerations. This method allowed estimation of all the mathematical parameters in the models. The physicochemical parameters intervening in the adsorption process that the models present could be deduced directly from the experimental adsorption isotherms by numerical simulation. We determine the adequate model for each type of isotherm, which fixes by direct numerical simulation the monolayer, multilayer, or condensation character. New equations are discussed and results obtained are verified for experimental data from the literature. The new theoretical expressions that we have proposed, based on statistical physics treatment, are rather powerful to better understand and interpret the various five physical adsorption type isotherms at a microscopic level.     

Theory of Gradient Elution Liquid Chromatography with Linear Solvent Strength: Part 1. Migration and Elution Parameters of a Solute Band

Based on the previous theoretical developments most notably by Snyder, this report offers the most complete theoretical framework of gradient elution LC with linear solvent strength (LSS). All statements of the theory are formulated as explicit mathematical expressions. The physics of chromatography in general and of the LSS model in particular were used only to justify the most basic mathematical expressions of the framework. Everything else was obtained by means of verifiable mathematical transformations. The framework was used for derivation of the largest systematic collection of mathematical expressions describing migration and elution parameters of a solute band. Majority of these expressions are new. They include not only the elution parameters of a band, but also previously unknown migration parameters as functions of distance and time traveled by the band. The set of the band parameters in this report was chosen on the basis of the needs for the study of the peak width formation (part 2 of this series) and for detailed study of performance of gradient LC similar to that recently published for temperature-programed GC. As an illustration of the utility of several parameters considered here, a simple way of prediction of a possibility of the reversal of a solute elution order due to the change in the gradient steepness has been found.     

Curve-fitting of Fourier manipulated spectra comprising apodization, smoothing, derivation and deconvolution

We present a general method for curve-fitting Fourier manipulated spectra, comprising apodized, smoothed, derivatised and deconvoluted spectra. The analytical expressions of Fourier manipulated bands in the spectral domain, needed for the curve-fitting, are usually very complex or do not even exist; hence an accurate curve-fit of Fourier manipulated spectra becomes unfeasible. Our strategy is to construct both the model and their derivatives in the Fourier domain, where they have simple and general expressions, and then Fourier transform them back to the spectral domain. The first benefit of this approach is the accurate curve-fitting of Fourier deconvoluted spectra, a main step in the secondary structure estimation of proteins by FTIR spectroscopy.     

Non-linear least-squares fitting with microsoft excel solver and related routines in HPLC modelling of retention

Summary Two problems related to non-linear regression, the evaluation of the best set of fitting parameters and the reliability of the methods used for the estimation of the standard errors of these parameters, are examined. It is shown that a non-linear curve fitting routine, like the Microsoft Excel Solver, may give more than one solution for the same data set and a simple Monte Carlo routine is described for the evaluation of the bestfit. For standard errors, the reliability of two procedures based on the conventional curvature matrix method, four Jackknife techniques and the bootstrap method are examined by comparing their results to those obtained from a Monte Carlo simulation of the experimental data. It is shown that a fitting parameter may follow a nonnormal distribution when the equation to be fitted is complicated, even if the errors on the data are normally distributed. In this case only Monte Carlo methods of data simulation can give accurate information about the standard errors and the confidence intervals of these parameters.     

Some difficult problems still existing in the preparation and certification of CRMs

Differences between particle size measurements of CRMs by various methods are discussed and the importance of the reliability of such data for proper estimation of the homogeneity of the material is emphasized. On the basis of a very simple model, the dependence of the Ingamells' sampling constant on the average mass of a single particle of the material is derived, and theoretical predictions are compared with the experimental results. Various approaches to the certification of the candidate RMs are briefly reviewed. The merits of the approach being used in this laboratory to evaluate data obtained in the interlaboratory comparison, and to assign certified and information values, is discussed. The conclusions are supported by results obtained for selected trace elements by use of "definitive" (primary) and "very accurate" methods. Some observations on the unusual resistance of some biological materials to wet ashing and the resulting possibility of making analytical errors are mentioned.     

Some difficult problems still existing in the preparation and certification of CRMs

Differences between particle size measurements of CRMs by various methods are discussed and the importance of the reliability of such data for proper estimation of the homogeneity of the material is emphasized. On the basis of a very simple model, the dependence of the Ingamells’ sampling constant on the average mass of a single particle of the material is derived, and theoretical predictions are compared with the experimental results. Various approaches to the certification of the candidate RMs are briefly reviewed. The merits of the approach being used in this laboratory to evaluate data obtained in the interlaboratory comparison, and to assign certified and information values, is discussed. The conclusions are supported by results obtained for selected trace elements by use of “definitive” (primary) and “very accurate” methods. Some observations on the unusual resistance of some biological materials to wet ashing and the resulting possibility of making analytical errors are mentioned.     

Separation mechanism of nortriptyline and amytriptyline in RPLC

The single and the competitive equilibrium isotherms of nortriptyline and amytriptyline were acquired by frontal analysis (FA) on the C18- bonded discovery column, using a 28/72 (v/v) mixture of acetonitrile and water buffered with phosphate (20 mM, pH 2.70). The adsorption energy distributions (AED) of each compound were calculated from the raw adsorption data. Both the fitting of the adsorption data using multi-linear regression analysis and the AEDs are consistent with a trimodal isotherm model. The single-component isotherm data fit well to the tri-Langmuir isotherm model. The extension to a competitive two-component tri-Langmuir isotherm model based on the best parameters of the single-component isotherms does not account well for the breakthrough curves nor for the overloaded band profiles measured for mixtures of nortriptyline and amytriptyline. However, it was possible to derive adjusted parameters of a competitive tri-Langmuir model based on the fitting of the adsorption data obtained for these mixtures. A very good agreement was then found between the calculated and the experimental overloaded band profiles of all the mixtures injected.     

Rupture force analysis and the associated systematic errors in force spectroscopy by AFM

Force spectroscopy is a new and valuable tool in physical chemistry and biophysics. However, data analysis has yet to be standardized, hindering the advancement of the technique. In this article, treatment of the rupture forces is described in the framework of the Bell-Evans model, and the systematic errors associated with the tether effect for approaches that utilize the most probable, the median, and the mean rupture forces are compared. It is shown that significant systematic errors in the dissociation rate can result from nonlinear loading with polymeric tethers even if the apparent loading rate is used in the analysis. Analytical expressions for the systematic errors are provided for the most probable and median forces. The use of these expressions to correct the associated systematic errors is illustrated by the analysis of the measured rupture forces between single hexadecane molecules in water. It is noted that the measured distributions of rupture forces often contain high forces that are unaccounted for by theoretical models. Experimental data indicate that the most significant effect of the high forces "tail" is on the dissociation rate obtained from the median force analysis whereas the barrier width appears to be unaffected.     

The effect of influential data, model and method on the precision of univariate calibration

Building a calibration model with detection and quantification capabilities is identical to the task of building a regression model. Although commonly used by analysts, an application of the calibration model requires at first careful attention to the three components of the regression triplet (data, model, method), examining (a) the data quality of the proposed model; (b) the model quality; (c) the LS method to be used or a fulfillment of all least-squares assumptions. This paper summarizes these components, describes the effects of deviations from assumptions and considers the correction of such deviations: identifying influential points is the first step in least-squares model building, the calibration task depends on the regression model used, and finally the least squares LS method is based on assumptions of normality of errors, homoscedasticity, independence of errors, overly influential data points and independent variables being subject to error. When some assumptions are violated, the ordinary LS is inconvenient and robust M-estimates with the iterative method of reweighted least-squares must be used. The effects of influential points, heteroscedasticity and non-normality on the calibration precision limits are also elucidated. This paper also considers the proper construction of the statistical uncertainty expressed as confidence limits predicting an unknown concentration (or amount) value, and its dependence on the regression triplet. The authors' objectives were to provide a thorough treatment that includes pertinent references, consistent nomeclature, and related mathematical formulae to show by theory and illustrative examples those approaches best suited to typical problems in analytical chemistry. Two new algorithms, calibration and linear regression written in s-plus and enabling regression triplet analysis, the estimation of calibration precision limits, critical levels, detection limits and quantification limits with the statistical uncertainty of unknown concentrations, form the goal of this paper.     

Equivalency of Kinetic Schemes: Causes and an Analysis of Some Model Fitting Algorithms

The same experimental data can often be equally well described by multiple mathematically equivalent kinetic schemes. In the present work, we investigate several model‐fitting algorithms and their ability to distinguish between mechanisms and derive the correct kinetic parameters for several different reaction classes involving consecutive reactions. We have conducted numerical experiments using synthetic experimental data for six classes of consecutive reactions involving different combinations of first‐ and second‐order processes. The synthetic data mimic time‐dependent absorption data as would be obtained from spectroscopic investigations of chemical kinetic processes. The connections between mathematically equivalent solutions are investigated, and analytical expressions describing these connections are derived. Ten optimization algorithms based on nonlinear least squares methods are compared in terms of their computational cost and frequency of convergence to global solutions. Performance is discussed, and a preferred method is recommended. A response surface visualization technique of projecting five‐dimensional data onto the three‐dimensional search space of the minimal function values is developed.     

Use of the Maximum Likelihood Method for Composition Data Analysis in the Radical Copolymerization Terminal Model

A method based on the maximum likelihood principle has been used for the estimation of reactivity ratios in the terminal model of copolymerization from experimental composition data. This method assumes explicitly that all measured variables are subject to errors. If correct estimates of experimental errors are used the best values of parameters are obtained. In addition, this method can also be used for the copolymerization model validation and/or evaluation of the accuracy of the experimental data. The application of the method is illustrated using simulated data disturbed with random errors.     

Analysis of the statistical error in umbrella sampling simulations by umbrella integration

Umbrella sampling simulations, or biased molecular dynamics, can be used to calculate the free-energy change of a chemical reaction. We investigate the sources of different sampling errors and derive approximate expressions for the statistical errors when using harmonic restraints and umbrella integration analysis. This leads to generally applicable rules for the choice of the bias potential and the sampling parameters. Numerical results for simulations on an analytical model potential are presented for validation. While the derivations are based on umbrella integration analysis, the final error estimate is evaluated from the raw simulation data, and it may therefore be generally applicable as indicated by tests using the weighted histogram analysis method.     

Influence of electrolytes on the dynamic surface tension of ionic surfactant solutions: expanding and immobile interfaces

Here, we derive analytical asymptotic expressions for the dynamic surface tension of ionic surfactant solutions in the general case of nonstationary interfacial expansion. Because the diffusion layer is much wider than the electric double layer, the equations contain a small parameter. The resulting perturbation problem is singular and it is solved by means of the method of matched asymptotic expansions. The derived general expression for the dynamic surface tension is simplified for the special case of immobile interface and for the maximum bubble pressure method (MBPM). The case of stationary interfacial expansion is also considered. The effective diffusivity of the ionic surfactant essentially depends on the concentrations of surfactant and nonamphiphilic salt. To test the theory, the derived equations are applied to calculate the surfactant adsorption from MBPM experimental data. The results excellently agree with the adsorption determined independently from equilibrium surface-tension isotherms. The derived theoretical expressions could find application for interpreting data obtained by MBPM and other experimental methods for investigating interfacial dynamics.     

A new method of estimating monomer reactivity ratios in copolymerization by linear regression methods

Sequential gas-liquid chromotographic analysis of the reaction mixture throughout a copolymerization reaction in conjuction with the improved curve-fitting I (integrated form) method, which accounts for measurements errors in both variables, allows accurate estimation of the monomer reactivity ratios. In this article an alternative method is presented for estimating r values in copolymerization with linear regression only, which is especially suited to cases in which one or two of the r values is close to 1. In these cases the improved curve-fitting I method tends to converge slowly because of the numerical instability of the integrated copolymerization equation. The use of the new method is illustrated for the estimation of the r values for ethylene and vinyl acatate in benzene at 35 kg/cm² and 62°C. The linear regression method was also tried on other copolymerizations and the results are compared with those obtained from the improved curve-fitting I method. The limits of the applicability of the linear regression method were determined by simulated sequential sampling experiments. It appears that the new method is applicable when the product of the r values is between 0.001 and 2, provided both monomer conversions are large enough compared with the measurements error.     

极大似然估计法及其在混合酸电位滴定中的应用

本文提出了以氢离子活度为测量值的混合酸滴定的计算公式;介绍了一种非线性模型参数估计方法--极大似然估计法(MLE),并用于混合酸的滴定计算,获得了比最小二乘估计法(LS)更为满意的结果;用蒙特卡罗方法对MLE和LS的结果作了研究。     

Direct analysis of small-angle equatorial x-ray scattering from fibrous systems. II. Methods of extracting physical parameters describing the scattering system

Small-angle equatorial x-ray scattering from fibrous systems is often complicated by the effects of morphological organization and/or heterogeneities as well as by macromolecular structure and organization influencing this same region of scattering space. In previous work it was shown that closed form mathematical expressions could be obtained for the Patterson function, without prior assumptions about the scattering system, when the intensity data were purposely truncated. In this paper the use of these expressions to extract the physical parameters describing the scattering system is demonstrated by application to a model system. The expressions are also expanded to include lattice distortions and long range coherency. A second approach is presented for cases when the coherence is small, and finally the use of higher-resolution data is discussed.     

Forward flux sampling-type schemes for simulating rare events: efficiency analysis

We analyze the efficiency of several simulation methods which we have recently proposed for calculating rate constants for rare events in stochastic dynamical systems in or out of equilibrium. We derive analytical expressions for the computational cost of using these methods and for the statistical error in the final estimate of the rate constant for a given computational cost. These expressions can be used to determine which method to use for a given problem, to optimize the choice of parameters, and to evaluate the significance of the results obtained. We apply the expressions to the two-dimensional nonequilibrium rare event problem proposed by Maier and Stein [Phys. Rev. E 48, 931 (1993)]. For this problem, our analysis gives accurate quantitative predictions for the computational efficiency of the three methods.     

How to compare diffusion processes assessed by single-particle tracking and pulsed field gradient nuclear magnetic resonance

Heterogeneous diffusion processes occur in many different fields such as transport in living cells or diffusion in porous media. A characterization of the transport parameters of such processes can be achieved by ensemble-based methods, such as pulsed field gradient nuclear magnetic resonance (PFG NMR), or by trajectory-based methods obtained from single-particle tracking (SPT) experiments. In this paper, we study the general relationship between both methods and its application to heterogeneous systems. We derive analytical expressions for the distribution of diffusivities from SPT and further relate it to NMR spin-echo diffusion attenuation functions. To exemplify the applicability of this approach, we employ a well-established two-region exchange model, which has widely been used in the context of PFG NMR studies of multiphase systems subjected to interphase molecular exchange processes. This type of systems, which can also describe a layered liquid with layer-dependent self-diffusion coefficients, has also recently gained attention in SPT experiments. We reformulate the results of the two-region exchange model in terms of SPT-observables and compare its predictions to that obtained using the exact transformation which we derived.     

On the Statistical Calibration of Physical Models

We introduce a novel statistical calibration framework for physical models, relying on probabilistic embedding of model discrepancy error within the model. For clarity of illustration, we take the measurement errors out of consideration, calibrating a chemical model of interest with respect to a more detailed model, considered as “truth” for the present purpose. We employ Bayesian statistical methods for such model‐to‐model calibration and demonstrate their capabilities on simple synthetic models, leading to a well‐defined parameter estimation problem that employs approximate Bayesian computation. The method is then demonstrated on two case studies for calibration of kinetic rate parameters for methane air chemistry, where ignition time information from a detailed elementary‐step kinetic model is used to estimate rate coefficients of a simple chemical mechanism. We show that the calibrated model predictions fit the data and that uncertainty in these predictions is consistent in a mean‐square sense with the discrepancy from the detailed model data.