The basic flow equations of electrophysiology in the presence of chemical reactions: II. A practical application concerning the pH and voltage effects accompanying the diffusion of O2 through hemoglobin solution

Previous papers by F. M. Snell (Jour. Theor. Biol.,8, 469–479, 1965) and M. A. Fox and H. D. Landahl (Bull. Math. Biophysics,27, Spec. Issue, 183–190, 1965) have found that the formulation by previous authors for the oxygen flow rates through hemoglobin solution as a function of pressure determined by E. Hemmingsen and P. F. Scholander (Science,132, 1379–1381, 1960) did not give a satisfactory quantitative fit of the curve for constant pressure difference. The suggestion of Fox and Landahl that the Bohr effect involving the shift in acidity accompanying the oxidation of Hb should give rise to voltage and pH differences in oxyhemoglobin transport is examined in more detail. In this paper, the previous expressions for the total oxygen flow rate in terms of the end point concentrations are extended to include the effects of the electrical field. Estimates of the potential difference shows it to be negligible. A derivation of a voltage-pH relation shows that the Nernst relation does not apply and a negligible voltage difference does not preclude a pH shift which is the more probable explanation of the discrepancies observed. Several other predictions suitable for experimental testing are made.     

Stochastic models for chemical reactions: I. Theory of the unimolecular reaction process

A stochastic model for the basic unimolecular chemical reaction is derived. This model provides a mathematical basis, altogether missing in the current kinetic theory, for the analysis of inherent random fluctuations about the strict concentration-time course prescribed by the existing deterministic theory. Limits on the extent of the predicted inherent variability are obtained and compared with those usually expected purely on the basis of random experimental errors of extraneous origin (not associated with the mechanism of reaction). The results support the extrapolation to chemical systems of a principle of statistical inaccuracy for physical systems which has been called by E. Schroedinger “the Law of Physics.”     

Erythrocyte distribution in arterial blood flow: I. Basic equations and anisotropic effects

Equations are derived for the flow of an anisotropic fluid in a tube. It is argued that these provide a model for arterial blood flow. Particular attention is paid to the effect of radial differences in hematocrit. Sequels to this paper (Bull. Math. Biophysics,29, 565–574; forthcoming, 1967) will respectively demonstrate possible wall-directed forces on the erythrocyte and enlarge on the physiological consequence of hematocrit variations. The present article develops the basic equations and explores the possible role of anisotropic effects in blood flow.     

Diffusion and simultaneous chemical reactions: I. A method for solving the equations of some systems in which a fixed concentration exists at a boundary

Many solutions are available to the differential equations for systems consisting of a space region with a boundary at which the concentration is fixed, diffusion occurring across this boundary. A method is described for readily transforming these solutions into results for similar systems in which the diffusing substance is removed by a first-order reaction and also removed or produced at a rate which is expressible as a polynomial in the time variable. Subsidiary transformations and steady-state conditions are also discussed. An indication is given of biological applications of the results made available by this method.     

Boundary conditions on the plasma emitter surface in the presence of a particle counter flow: I. Ion emitter

Emission of positively charged ions from a plasma emitter irradiated by a counterpropagating electron beam is studied theoretically. A bipolar diode with a plasma emitter in which the ion temperature is lower than the electron temperature and the counter electron flow is extracted from the ion collector is calculated in the one-dimensional model. An analog of Bohm’s criterion for ion emission in the presence of a counterpropagating electron beam is derived. The limiting density of the counterpropagating beam in a bipolar diode operating in the space-charge-limited-emission regime is calculated. The full set of boundary conditions on the plasma emitter surface that are required for operation of the high-current optics module in numerical codes used to simulate charged particle sources is formulated.     

Diffusion and simultaneous chemical reactions: II. The equations of those systems in which transport occurs from one region to an adjoining region

A method is described for obtaining the solutions to the equations of systems in which a reacting substance diffuses between two adjacent regions. This substance may be produced or removed in the two regions at rates expressible as polynomial functions of time and may be removed throughout both regions by a first-order reaction. The solutions are obtained from simpler results, many of which are available and more of which are listed in this paper. Possible application in the study of the validity of cytochemical staining procedures is discussed.     

2-aminopropionitrile polymer. I. The hydrolyzate of the basic fraction

The basic fraction of a 2-aminopropionitrile polymer was subjected to acid hydrolysis and was analyzed by means of GC-MS, after trimethylsilylation. The fundamental polymer structural units were alanine, 2,2-iminodipropionic acid, N-(1-cyanoethyl)alanine, N-ethylalanine, glycine, cyanoglycine, and 5-amino-4-carboxyimidazole residues. The last three units may be derived from hydrogen cyanide. Oligomeric combinations of these units were also detected in the hydrolyzate, due to partial hydrolysis of the polymer.     

Determination of three different pools of reduced one-carbon-substituted folates: I. A study of the fundamental chemical reactions

The reduced one-carbon-substituted derivatives of folic acid can be grouped in three pools according to their response to acid treatment. Pool 1 is made up of N⁵,N¹⁰-methylene-tetrahydrofolic acid and unsubstituted dihydro- and tetrahydrofolic acid which at pH 1.0 and subsequent exposure to air cleave to p-aminobenzoylglutamic acid. Pool 2 is made up by the acid-stable N⁵-methyl-tetrahydrofolic acid, and pool 3 includes N⁵,N¹⁰-methenyl-tetrahydrofolic acid, N¹⁰-formyltetrahydrofolic acid, N⁵-formyltetrahydrofolic acid, and N⁵-formiminotetrahydrofolic acid, all of which convert to the stable N⁵,N¹⁰-methenyl-tetrahydro form when acid treated. Conditions are described to selectively cleave the C⁹-N¹⁰ bond of the folates of pool 1, pools 1 + 2, and pools 1 + 2 + 3. The cleaved pools are quantitated as the Bratton-Marshall azo dyes of p-aminobenzoylglutamate. The uncleaved pools are converted to Bratton-Marshall-negative products. Pool 1 is determined by converting pool 2 to 4a-hydroxy-5-methyltetrahydrofolic acid and pool 3 to N¹⁰-formylfolic acid, both Bratton-Marshall negative, by 10% hydrogen peroxide oxidation at pH 6.0. Pools 1 + 2 are cleaved with 0.015% hydrogen peroxide and 0.1% potassium permanganate at pH 9.0 which convert the N⁵-methyltetrahydrofolic acid to the acid-cleavable N⁵-methyl-dihydrofolic acid. Pool 3 oxidizes to the Bratton-Marshall-negative N¹⁰-formylfolic acid. Pools 1 + 2 + 3 are cleaved by first reducing pool 3 to N⁵-methyltetrahydrofolic acid with sodium borohydride followed by oxidation at pH 9.0 to its acid-labile dihydro form. Determination of the poly-γ-glutamyl chain length of each pool is possible by chromatographing the azo-p-aminobenzoylpolyglutamates with authentic synthetic markers.     

Stochastic models for chemical reactions: II. The unimolecular rate constant

Based on the author's stochastic treatment of the unimolecular reaction which contains a mathematical rationale for considering random fluctuations inherent in the reaction mechanism (Bartholomay, 1958,Bull. Math. Biophysics,20, 175-90), a new and simple formula is derived for calculating the rate constants of such reactions. This formula is then applied to data from the literature and the results compared with estimates obtained alternatively using current methods which attribute all irregularities about the usually expected smooth, decaying exponential curve to extraneous causes unrelated to reaction mechanism. Both estimates are seen to be almost identical despite the opposing assumptions about the origin of the random component of the kinetic data of such reactions.     

Discussion on basic equations for a flow in human air passages and their representation

Basic equations of time-dependent flow are discussed in the case of rigid or elastic branches and according to the airway representation pattern, both continuous and discontinuous. Two relevant fundamental continuous functions are defined: the ‘order of generation’ G(x) and the cumulative bronchial diameter D(x). A numerical solution is outlined in the former case, then in the latter case, and the system of equations which completely describes the flow is solved.