Derivation of unstirred-layer transport number equations from the Nernst-Planck flux equations.

Since the late 1960s it has been known that the passage of current across a membrane can give rise to local changes in salt concentration in unstirred layers or regions adjacent to that membrane, which in turn give rise to the development of slow transient diffusion potentials and osmotic flows across those membranes. These effects have been successfully explained in terms of transport number discontinuities at the membrane-solution interface, the transport number of an ion reflecting the proportion of current carried by that ion. Using the standard definitions for transport numbers and the regular diffusion equations, these polarization or transport number effects have been analyzed and modeled in a number of papers. Recently, the validity of these equations has been questioned. This paper has demonstrated that, by going back to the Nernst-Planck flux equations, exactly the same resultant equations can be derived and therefore that the equations derived directly from the transport number definitions and standard diffusion equations are indeed valid.     

Virial Expansions for Ideal Self-Associating Systems

Theoretical virial equations for self-associating systems governed by mass action have been derived assuming the solute to be ideal except for this solutesolute interaction. In particular, monomer-polymer association involving two molecular species and isodesmic association involving an indefinite number of molecular species have been treated analytically. The usefulness of such virial equations is severely limited by their extremely narrow interval of convergence.     

Kinetics of a model for zymogen activation: the case of high activating enzyme concentrations

Transient phase and steady state equations have been derived for the following enzyme activation mechanism: (formula; see text) in which the concentrations of activating enzyme, E, and substrate, A, are greatly in excess of that of zymogen, Ei. EEi, EEa, EaA and EaY are four intermediates; W is a peptide related from Ei during EEa formation and X and Y are the products of Ea reaction on A. From the general equations, approximate solutions under certain simplifying conditions have been derived. Finally, some formal particular cases of the above mechanism are considered.     

DNA的振动—内激发—转子模型

本文全面考察了DNA分子链的各种运动自由度,从B型DNA的双螺旋结构出发,建立了DNA的振动动-内激发-转子模型,并由此给出了相应的哈密顿函数、运动方程和孤子解,较为全面地阐明了DNA的动力学特性．     

THE THEORY OF DIFFUSION IN CELL MODELS

The differential equations which describe the simultaneous diffusion of water and a salt in a cell model have been formulated and solved. The equations have been derived from the general laws which describe diffusion processes, thereby furnishing a physical interpretation for the constants which enter into the theory. The theoretical time curves for the two diffusing substances are in good agreement with the experimentally determined curves and accurately reproduce all of the essential characteristics of the experiment.     

The Replication of DNA: II. The Number of Polynucleotide Strands in the Conserved Unit of DNA

The kinetics of degradation of DNA by deoxyribonuclease II have been studied, using the techniques of light scattering, viscosity, and titration. Theoretical equations have been derived for both random and non-random attacks, and all assumptions have been evaluated. It has been shown that these equations permit a valid calculation of the number of polynucleotide strands per molecule. The results have been verified by two independent experimental methods. DNA from proliferating sources was found to be four-stranded; DNA from non-proliferating sources was found to be two-stranded. The implications of these findings are discussed.     

AUTOTRIPLOID AND AUTOTETRAPLOID CYTOGENETIC ANALYSES: CORRECTION COEFFICIENTS FOR PROPOSED BINOMIAL MODELS

Correction coefficients have been derived for two-chiasmata models of autotriploids and autotetraploids to correct meiotic configuration frequencies obtained from the binomial method when the P values are 0.5. In addition, equations are derived for the direct calculation of meiotic configuration frequencies in autotriploids of the two-chiasmata model when P values are 0.5.     

On the relationship between response time and dosage of a drug as a function of its mode of entry

Theoretical studies have been carried out to determine the effect of the route of entry and the temporal sequence of dosage on the response. On the basis of continuity equations applied to a schematized system, equations relating the dosage and response time have been derived for some special cases, and the form of this relatinship is discussed. This paper is based in part on work done for the Office of Scientific Research and Development under Contract NDCrc-132 with the University of Chicago.     

On the Equivalence of Mathematical Models For Cell Proliferation Kinetics

Various types of mathematical models, such as partial differential equations, ordinary differential equations and difference equations, are available in the literature to describe the kinetics of cell proliferation, and different studies of cell kinetic phenomena have been conducted using these models. This paper discusses the equivalence between the different models identifying the conditions and approximations under which one type of models may be derived from another. Such an equivalence study is highly useful for an integration of the diverse results that have been obtained using different models in order to gain a more complete understanding of cell kinetic phenomena.     

Bacteriorhodopsin in liposomes. II. Experimental evidence in support of a theoretical model.

In the preceding article equations describing relevant ion flows in illuminated suspensions of bacteriorhodopsin liposomes have been derived. Here these equations are subjected to experimental tests. Changes in permeability characteristics of the liposomal membrane are brought about by addition of specific ionophores and change of medium composition. Using light-driven proton uptake and electrochemical potential differences for protons across the membrane as observation parameters, ridig attempts to falsify the derived equations are unsuccessful. Agreement between equations and experimental results is established on the point of: (i) the antagonistic effect of valinomycin and nigericin on the two components of the proton-motive force, (ii) the time dependence of the changes in transmembrane electrical and chemical potential differences after the onset of illumination. In three independent experimental systems evidence was obtained for the correctness of the postulated dependence of the turnover rate of the photochemical cycle on back pressure by the transmembrane electrochemical potential difference for protons.     

Bacteriorhodopsin in liposomes. II. Experimental evidence in support of a theoretical model

In the preceding article equations describing relevant ion flows in illuminated suspensions of bacteriorhodopsin liposomes have been derived. Here these equations are subjected to experimental tests. Changes in permeability characteristics of the liposomal membrane are brought about by addition of specific ionophores and change of medium composition. Using light-driven proton uptake and electrochemical potential differences for protons across the membrane as observation parameters, ridig attempts to falsify the derived equations are unsuccessful.Agreement between equations and experimental results is established on the point of: (i) the antagonistic effect of valinomycin and nigericin on the two components of the proton-motive force, (ii) the time dependence of the changes in transmembrane electrical and chemical potential differences after the onset of illumination.In three independent experimental systems evidence was obtained for the correctness of the postulated dependence of the turnover rate of the photochemical cycle on back pressure by the transmembrane electrochemical potential difference for protons.     

The problem of exchange between two or more individuals,motivated by hedonistic considerations

Let two or more individuals each possess different quantities of two objects of satisfaction. Under certain conditions they may agree to exchange part of the objects if this leads to an increase of each one's satisfaction. The equations which govern this process have been derived by G. E. Evans (1930) for the case of two individuals. A different proof of these equations is given here and the equations are generalized to the case of more than two individuals.     

SWELLING OF ERYTHROCYTES IN SOLUTIONS OF AMMONIUM SALTS

Two rather simple equations have been derived, which make it possible to express in a single number the result of a series of determinations of the volume of erythrocytes swelling in solutions of ammonium salts. In all experiments made with several combinations of different concentrations of permeating and non-permeating salts, the curves calculated from the equations have covered the points found by experiment.     

Liquid membrane potential in nonisothermal systems.

Electrical membrane potential equations for liquid ion exchange membranes, characterized by the presence of uncharged associated species and by exclusion of co-ions (no electrolyte uptake) have been derived. The irreversible thermodynamic theories already developed for solid membranes with fixed charged site density have been extended to include the different physicochemical aspects of the liquid membranes. To this purpose the dissipation function has been written with reference to the fluxes of all the species present in the membrane. It has been found that the mobile charged site, the counterions, and the uncharged associated species contribute to the electrical membrane potential through their phenomenological coefficients. The electrical membrane potential equations have been integrated in isothermal and nonisothermal conditions for monoionic and biionic systems. The theoretical predictions have been experimentally tested by studying the electrical potential of liquid membranes formed with solutions of tetraheptylammonium salts in omicron-dichlorobenzene.     

Deriving the rate equations for product inhibition patterns in bisubstrate enzyme reactions

In this work, the full rate equations for 17 completely reversible bisubstrate enzyme kinetic mechanisms, with two substrates in the forward and two in the reverse direction, have been presented; among these are rapid equilibrium, steady-state, and mixed steady-state and rapid equilibrium mechanisms. From each rate equation eight product inhibition equations were derived, four for the forward and four for the reverse direction. All the corresponding product inhibition equations were derived in full; thus a total of 17 × 8 = 136 equations, were presented. From these equations a list of product inhibition patterns were constructed and presented in a tabular form, both for the primary plots (intercept effects) and the secondary plots (slope effects).

The purpose of this work is to help investigators in practical work, especially biologists working with enzymes, to choose quickly an appropriate product inhibition pattern for the identification of the kinetic mechanism. The practical application of above product inhibition analysis was illustrated with three examples of yeast alcohol dehydrogenase-catalyzed reactions.     

Deriving the rate equations for product inhibition patterns in bisubstrate enzyme reactions

In this work, the full rate equations for 17 completely reversible bisubstrate enzyme kinetic mechanisms, with two substrates in the forward and two in the reverse direction, have been presented; among these are rapid equilibrium, steady-state, and mixed steady-state and rapid equilibrium mechanisms. From each rate equation eight product inhibition equations were derived, four for the forward and four for the reverse direction. All the corresponding product inhibition equations were derived in full; thus a total of 17 x 8 = 136 equations, were presented. From these equations a list of product inhibition patterns were constructed and presented in a tabular form, both for the primary plots (intercept effects) and the secondary plots (slope effects). The purpose of this work is to help investigators in practical work, especially biologists working with enzymes, to choose quickly an appropriate product inhibition pattern for the identification of the kinetic mechanism. The practical application of above product inhibition analysis was illustrated with three examples of yeast alcohol dehydrogenase-catalyzed reactions.     

The chequered history of the development and use of simultaneous equations for the accurate determination of chlorophylls a and b

Over the last half century, the most frequently used assay for chlorophylls in higher plants and green algae, the Arnon assay [Arnon DI (1949) Plant Physiol 24: 1–15], employed simultaneous equations for determining the concentrations of chlorophylls a and b in aqueous 80% acetone extracts of chlorophyllous plant and algal materials. These equations, however, were developed using extinction coefficients for chlorophylls a and b derived from early inaccurate spectrophotometric data. Thus, Arnon's equations give inaccurate chlorophyll a and b determinations and, therefore, inaccurate chlorophyll a/b ratios, which are always low. This paper describes how the ratios are increasingly and alarmingly low as the proportion of chlorophyll a increases. Accurate extinction coefficients for chlorophylls a and b, and the more reliable simultaneous equations derived from them, have been published subsequently by many research groups; these new post-Arnon equations, however, have been ignored by many researchers. This Minireview records the history of the development of accurate simultaneous equations and some difficulties and anomalies arising from the retention of Arnon's seriously flawed equations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Proportionality of Diagonal Blocks with Blockwise Independence

In this paper, the tests of similarities among group covariance matrices and the differences among block covariance matrices within a group under repeated measurement model are studied. There are nine hierarchical nested structures of covariance matrices which have been tested. The likelihood ratio tests have been derived for these nine hierarchically structured models. An algorithm for determining the numerical solution of the corresponding maximum likelihood equations is also given.     

Kinetics of suicide substrates.

When suicide substrates inactivate enzymes during catalysis, formation of product and inactivation of enzyme proceed concurrently. The steady-state hypothesis is applicable when catalytic quantities of enzyme are used. Equations for the rate of inactivation have been derived and integrated to obtain equations describing progress curves.     

The Role of Membrane Carriers in the Regulation of the Free Glucose Pool in Metabolizing Cells

SYNOPSIS. Equations have been derived which relate the intracellular concentration of a metabolized sugar to its extracellular concentration and the kinetic constants for transport and phosphorylation. The equations show that the maximum intracellular, concentration of a metabolized sugar is a constant determined by the kinetic constants for transport and phosphorylation.