Statistical data processing of mobility curves of univalent weak bases

This paper brings a new rigorous and complete statistical approach to the data processing of the mobility curves of univalent weak bases. This approach is based on application of the least square method to the equation of the related mobility curve. Thus, an equation for the best fit is derived and its mathematical solution is found. The solution brings best estimates of the mobility curve parameters, i.e., dissociation constant K and ionic mobility of the protonated base U. Further, explicit formulas have been derived for the calculation of related statistical parameters, i.e., SDs of effective mobility s(u), of the dissociation constant s(K), and of ionic mobility of protonated base s(U). The mathematical functions used in the above approach do not impose any limitations on the data used, i.e., the mobility and pH values used may be real numbers (positive, negative, zero).     

Modified Booth equation for the calculation of zeta potential

 An easy-to-use computer program based on the modified Booth equation (MBE) is developed to calculate the zeta potential of a spherical nonconducting particle from knowledge of the electrophoretic mobility, particle size, and the type and concentration of ions present in the solution. The program is applied to five sets of previously published literature data and the resulting zeta potentials are compared with the values given by the Henry equation to illustrate the extent of the relaxation effect in each case. In four cases, the output zeta potential data are compared with the corresponding values obtained from the rigorous numerical solution of O’Brien and White. Results indicate that the computer program developed here gives a reliable estimate of the zeta potential. The main advantage in using the MBE lies in its capability of calculating the zeta potential even for cases where the experimentally measured mobility exceeds the maximum theoretical mobility predicted by the O’Brien and White solution. Received: 16 June 1997 Accepted: 11 September 1997     

Statistical and Mathematical Investigations for Determining the Binding Constants of Micelle with Reactants Molecules from Kinetic Data

Binding constants between reactants molecules with micelles are considered to be important parameters particularly in micellar catalysis area. Recently, we developed a statistical method based on multiple linear regression for determining those parameters from kinetic data (Phys. Chem. Liq. 2008, 46, 34–46). In the present work, we derived further two statistical equations from the same original equation using also multiple linear regression method. A substantial difference has been found between the results of those equations and with that of the recently published one. This strongly indicates that the statistical procedures are not valid for such a purpose, that is, the available statistical and graphical methods in the literature are also not suitable for such treatment. A mathematical procedure using iterative method for evaluating the binding constants is introduced. An equation for such treatment has also been derived from the same original equation, and a computer program for this purpose has been written. Application of the developed method to the kinetic data has been found to be quite successful. It has been concluded that the presented mathematical method is simple, reliable, and accurate.     

Effect of mobile phase amine additives on enantioselectivity for phenylalanine analogs

Curve fitting seems to be one of the best methods for the evaluation of chromatographic signals. As it is known, in this case mathematical function is fitted to digitized measured points. The most important task is to find the best mathematical function, which corresponds perfectly to the peak shape, and then to determine the parameters of the equation using a computerized least-squares method of approximation. In this work, a new mathematical function was sought for with the purpose of describing different chromatographic signals and it was fitted to the digitized measured points. The fitted curve is suitable for a quick evaluation of chromatographic information, noise filtering and correction of baseline drift. The fitting of gas chromatographic and high-performance liquid chromatographic signals were completed. The mathematical function, the generated chromatographic curves, the application of the function for describing real signals and the fitting process will be demonstrated in this study.     

Evaluation of optimization techniques for an extractive alcoholic fermentation process

The mathematical optimization of a continuous alcoholic fermentation process combined with a flash column under vacuum was studied. The objective was to maximize % yield and productivity in the fermentor. The results using surface response analysis combined with modeling and simulation were compared withy those obtained when the problem was written as a nonlinear programming problem and was solved with a successive quadratic programming (SQP) technique. Two process models were evaluated when the process was optimized using the SQP technique. The first one is a deterministic model, whose kinetic parameters were experimentally determined as functions of the temperature, and the second is a statistical model obtained using the factorial design technique combined with simulation. Although the best result was the one obtained using the rigorous model, the values for productivity and % yield obtained using the simplified model are acceptable, and these models can be used when the development of a rigorous model is excessively difficult, slow, or expensive.     

Thermodynamic equations for the isobaric and isochoric heat capacities of pure substances at the critical point

It was proved on the assumption of a nonzero and finite slope of the elasticity curve at the critical point that the isochoric heat capacity at this point cannot be established on the basis of thermodynamics only. The critical conditions of a pure substance were derived from the differential equations of thermodynamics using a rigorous mathematical apparatus. Several indeterminate forms containing isochoric or isobaric heat capacities or their derivatives were evaluated.     

Kinetic approach to partially overlapped thermal decomposition processes

Practical usefulness of the kinetic deconvolution for partially overlapped thermal decomposition processes of solids was examined by applying to the co-precipitated basic zinc carbonate and zinc carbonate. Comparing with the experimental deconvolutions by thermoanalytical techniques and mathematical deconvolutions using different statistical fitting functions, performance of the kinetic deconvolution based on an accumulative kinetic equation for the independent processes overlapped partially was evaluated in views of the peak deconvolution and kinetic evaluation. Two-independent kinetic processes of thermal decompositions of basic zinc carbonate and zinc carbonate were successfully deconvoluted by means of the thermoanalytical measurements in flowing CO₂ and by applying sample controlled thermal analysis (SCTA). The deconvolutions by the mathematical curve fittings using different fitting functions and subsequent formal kinetic analysis provide acceptable values of the mass-loss fractions and apparent activation energies of the respective reaction processes, but the estimated kinetic model function changes depending on the fitting functions employed for the peak deconvolution. The mass-loss fractions and apparent kinetic parameters of the respective reaction processes can be optimized simultaneously by the kinetic deconvolution based on the kinetic equation through nonlinear least square analysis, where all the parameters indicated acceptable correspondences to those estimated through the experimental and mathematical deconvolutions. As long as the reaction processes overlapped are independent kinetically, the simple and rapid procedure of kinetic deconvolution is useful as a tool for characterizing the partially overlapped kinetic processes of the thermal decomposition of solids.     

Use of a polynomial exponential function to describe migration of proteins on sodium dodecyl sulfate-polyacrylamide gels

Standard mixtures of proteins were separated by sodium dodecyl sulfate-polyacrylamide gel electrophoresis. Polynomial regression analysis was used to fit curves to the data points obtained by plotting log10 of protein molecular weight versus electrophoretic mobility. Polynomials with orders ranging from 1 to 4 were generated. The coefficients of each equation were analyzed for statistical significance. It was found that a third order polynomial was the highest-order equation in which all coefficients contributed significantly to the prediction of molecular weights. Using this equation, it was possible to estimate the molecular weights of known proteins in the range from 97,400 to 14,400 with a maximum error of 1%, compared with a maximum error of 17% when a first-order equation was used to describe the migration of the standards.     

一种改进的休克尔分子轨道模型预测纯碳环分子稳定结构

从动力学和统计热力学角度介绍了BET方程中C的物理意义,分析了C的大小对吸附等温线形状的影响,推导了吸附等温线上拐点、B点位置和C大小的关系以及达到单程吸附量时C与相对压力、覆盖表面分数的关系。文章用数学方法介绍了保证C为正值的方法,解释了数据分析中的一个常见现象,还从BET方程的数学形式及物理意义探讨了C的下限。     

The statistics of ToF‐SIMS data revisited and introduction of the empirical Poisson correction

Generation of time‐of‐flight secondary ion mass spectrometry (ToF‐SIMS) data involves two overarching processes: secondary ion production and secondary ion detection. The interpretation of ToF‐SIMS data is facilitated if the intensities of the as‐measured mass spectra are proportional to the abundances of the species under investigation. While secondary ion yield is normally taken to be a linear process, ion detection is not owing to detector dead‐time effects. Consequently, methods have been devised that attempt to linearize, or correct, data that are affected by the dead time. In this article, we review the statistics of ToF‐SIMS data generation and confirm a report in the literature that abundance estimates from so‐called Poisson corrections are biased. We show that these corrections are only unbiased asymptotically and that a rigorous probabilistic analysis can quantitatively account for the observed bias. Two sources of bias are identified, one having a statistical basis and one due to the form of the correction equation at high ion detection rates. Based on insights gained from this analysis, we propose a new correction equation, the empirical Poisson correction, which largely eliminates the statistical bias. The performance of the proposed correction is illustrated by reanalyzing 14 experimentally measured datasets that suffer from varying levels of dead‐time effects. Copyright © 2016 John Wiley & Sons, Ltd.     

电渗流与缓冲液浓度之间数学关系的线性表达

It was believed that electroosmotic mobility μeo is inversely proportional to the square root of the ionic strength L But the linear relationship for regression analysis was expressed differently in different papers. The paper studied the linear expression of the mathematical relationship between μeo and c （background buffer concentration） by mathematical transform and real experimental data.μeo values of fused silica capillary were determined in four buffer systems. Their experimental conditions were controlled carefully for decreasing temperature difference AT and pH difference ApH in 50 μm ID capillary, in which no double layer overlap existed. The linear relationship between the reciprocal of electroosmotic mobility and the square root of concentration （or ionic strength） was derived by mathematical method. The regression analysis of experimental data was shown to well correspond to the relationship. The constants in regression equation could be well defined and the calculated results were acceptable.     

小电流交流示波极谱E～t曲线理论公式的推导方法

本文用电极等效电路原理推导了小交流电流情况下的E～t曲线方程式,在近似应用于大电流的情况时,与Micka公式相似,当进一步近似时即可得Micka公式,但在纯充电时可得到与Heyovsky公式相一致的结果,弥补丁Micka公式的不足,能说明一些Micka公式不能说明的现象。     

Assessment of uncertainty in parameter evaluation and prediction

Like in all experimental science, chemical data is affected by the limited precision of the measurement process. Quality control and traceability of experimental data require suitable approaches to express properly the degree of uncertainty. Noise and bias are nuisance effects reducing the information extractable from experimental data. However, because of the complexity of the numerical data evaluation in many chemical fields, often mean values from data analysis, e.g. multi-parametric curve fitting, are reported only. Relevant information on the interpretation limits, e.g. standard deviations or confidence limits, are either omitted or estimated. Modern techniques for handling of uncertainty in both parameter evaluation and prediction are strongly based on the calculation power of computers. Advantageously, computer-intensive methods like Monte Carlo resampling and Latin Hypercube sampling do not require sophisticated and often unavailable mathematical treatment. The statistical concepts are introduced. Applications of some computer-intensive statistical techniques to chemical problems are demonstrated.     

Finding reaction paths using the potential energy as reaction coordinate

The intrinsic reaction coordinate curve (IRC), normally proposed as a representation of a reaction path, is parametrized as a function of the potential energy rather than the arc-length. This change in the parametrization of the curve implies that the values of the energy of the potential energy surface points, where the IRC curve is located, play the role of reaction coordinate. We use Caratheodory's relation to derive in a rigorous manner the proposed parametrization of the IRC path. Since this Caratheodory's relation is the basis of the theory of calculus of variations, then this fact permits to reformulate the IRC model from this mathematical theory. In this mathematical theory, the character of the variational solution (either maximum or minimum) is given through the Weierstrass E-function. As proposed by Crehuet and Bofill [J. Chem. Phys. 122, 234105 (2005)], we use the minimization of the Weierstrass E-function, as a function of the potential energy, to locate an IRC path between two minima from an arbitrary curve on the potential energy surface, and then join these two minima. We also prove, from the analysis of the Weierstrass E-function, the mathematical bases for the algorithms proposed to locate the IRC path. The proposed algorithm is applied to a set of examples. Finally, the algorithm is used to locate a discontinuous, or broken, IRC path, namely, when the path connects two first order saddle points through a valley-ridged inflection point.     

Computerized methods for analyzing two-dimensional agarose gel electropherograms

Previous methods interpret zonal or polydisperse gel patterns of two-dimensional Serwer-type gels in terms of size and free mobility (surface net charge density). These two parameters have been determined for each component without quantitatively measuring the abundance of the components. The present study advances these previous methods by determining the relative concentration of each component by computer evaluation of densitometrically analyzed gel patterns. Suitable procedures and their underlying algorithms are presented. The mathematical routines are implemented in a user-friendly software package, called GelFit and designed for a Macintosh personal computer. The program input consists of digitized images of gel staining patterns exemplified by those obtained from electrophoresis of native subcellular-sized particles. The data are processed through the following steps: (i) Noise reduction and calibration. (ii) Geometrical transformation of the pattern onto a rectangular size/free mobility coordinate system using rationales of the extended Ogston model. (iii) Analysis of the transformed image to determine density maxima, density profiles along iso-free-mobility or iso-size lines, curve fitting of one-dimensional profiles or two-dimensional surfaces using Gaussian functions and curve stripping of surfaces to determine the possible number of particle populations.     

Mathematical Modelling and Computer Simulation of Linear Polymer Degradation: Simple Scissions

Summary: Degradation of a polymer in a reactor by the degrading agent(s) follows a distinct pattern, primarily influenced by structural integrity and reactor environment. This distinct pattern is recorded in the changes in the evolved molecular weight distribution (MWD) or polymer chain length distribution (PCLD) curve characteristics from the initial intact state. Modern size exclusion chromatography (SEC) is the best laboratory‐based method that can clearly provide these plots in the form of chromatogram; however, detailed molecular information is not available. The nature of molecular destruction can be well‐characterised if the distinct MWD shift patterns can be simulated to fingerprint the different chain scission dynamics. This is investigated by our current research using the power of computer simulation techniques to gain insight into the polymer ageing processes. One such technique for studying simple decay processes is presented here, and the results are compared with experimental findings. The concept of a binary tree scission model is introduced to show chain rupture as a sequence of probabilistic events and as a non‐linear function of time. Two new mathematical algorithms, an iterative Monte Carlo structured probability scheme and a semi‐iterative algebraic exact statistical formulation method, are investigated to implement this model and simulate the evolution of resultant temporal MW distribution. The latter, an innovative approach to mathematical modelling, has the potential to generate a statistically perfect instant MWD decay curve. A statistical comparison of the product yield is presented from the data obtained using a wide variety of simulated scission regimes to determine the sources of variability.

Simulated MWD lateral shift for percent cut scission model showing deviation from the initial MWD (red) over degradation time zones T_j(0 ≥ j ≤ 9) with bimodal and curve broadening effect due to accumulation of varied percent cut range 5–30%.     

Electrophoretic mobility equation for protein with molecular shape and charge multipole effects

We derive a simple formula for the free solution electrophoretic mobility of protein by including both molecular shape and charge distribution effects. The molecular shape of protein is described by a deformed sphere model, while the charge distribution is represented in terms of net charge, charge dipole, and charge quadrupole. The deformed sphere model approximates the radial coordinate of the protein surface as a simple quadratic equation based on the atomic coordinate data. Charge dipole does not affect the mobility of protein. Combined with the quadratic coefficients of the surface equation, charge quadrupole affects the mobility. When the charge quadrupole contribution is negligible, the mobility equation simplifies to the Henry equation in which the sphere radius is replaced with the hydrodynamic radius of protein. The deformed sphere model predicts correctly the hydrodynamic radius of protein from the atomic coordinate data. The hydrodynamic radius is not the radius of sphere of equal volume but the effective radius that correlates with the translational diffusivity of protein. To illustrate the utility of our mobility equation we study the electrophoresis of lysozyme and compare our results with previously published works.     

复杂化学体系测量数据的主因子数估计方法的分类及进展

复杂化学体系的数据解析中,估计主因子数是必要步骤,同时也是一个难题.目前存在多种各具特点的估计方法.对这些方法进行整理和归纳可为解决这一难题提供全局视角下的综合性信息,有助于进一步研究.本文结合本课题组的相关研究对主因子数估计问题进行了系统深入的分析;整理了近年来出现的各种估计方法,并将之分为三类,即经验方法、数学原理完备的方法以及统计方法,并对每类方法的特点和共性进行了分析和说明;归纳出第二类方法的基本原理.