George Karreman

Electrical potentials and ionic fluxes in ion exchangers: I. “n type” non-ideal systems with zero current

In this paper a derivation, based on the Nernst-Planck flux equations extended for non-ideal behavior, is given for the electrical potentials and cationic fluxes occurring across a one-phase ion exchange system separating two aqueous solutions. It is assumed that the activities of the cations in the exchanger are proportional to thenth power of their concentrations, a relationship of broad empirical validity. This paper deals only with the case of the quasi-stationary state in which no net electrical current flows. The profles of the cationic concentrations and of the diffusion potential in the exchanger are found to be, in general, exponential and linear respectively. An equation is derived for the stationary-state potential of an ion exchange phase interposed between two aqueous solutions containing mixtures of two cations. The potential selectivity constant of this equation is found to equal the product of the ion exchange equilibrium constant and thenth power of the mobility ratio.

Some contributions to the mathematical biology of blood circulation. Reflections of pressure waves in the arterial system

If, along the length of a blood vessel, there is a discontinuous change of diameter, the pressure waves will be reflected at the discontinuity. The transmission and reflection coefficients of the waves are calculated in terms of the ratio of the diameters and the ratio of the propagation velocities of the waves in the parts of the blood vessel. We obtain also the reflection coefficient in terms of the ratios of the diameters, of the elasticity moduli, and of the wall thicknesses.

Contributions to quantum biology. I. Mobile electronic characteristics of riboflavin radicals

Quantum biology is the quantum mechanical study of electrons in molecules of biological interest. This requires the solution of problems involving many electrons. Approximation methods are therefore necessary and are discussed. The present study, concerned with the mobile electrons in riboflavin (FMN) and its radicals (FMN−, FMNH and FMNH2+), is based on the approximation method, developed by B. Pullman and A. Pullman. The solution of the eigenvalue problems so obtained gives the energy levels of the mobile electron systems involved. The corresponding eigenvectors yield the mobile electronic charges of the atoms of riboflavin radicals which have contributed mobile electrons. Important differences of the net charge distributions of these radicals are emphasized. The longest wave length of light absorption is calculated from the obtained energy levels and agrees, within the accuracy of the method, with corresponding experimental results. From the appropriate calculated results, electronic assignments are obtained for the experimental transitions involved.

Contributions to the mathematical biology of excitation with particular emphasis on changes in membrane permeability and on threshold phenomena

Simple chemical reactions of Ca++ and K+ ions with a P− ion in a membrane are assumed to be causes of changes in permeability of that membrane to K+ ions. On the basis of such a mechanism a quantative concept of membrane permeability to K+ ions is defined. The diffusion of K+ ions through such a membrane is studied mathematically in a simplified version. An applied electrical field as well as the diffusion potential of the K+ ions are considered to effect the chemical equilibrium constants of the proposed reactions. It is shown that such a system, which can be described by a set of nonlinear differential equations, may have two stable states of equilibrium which are separated by an unstable equilibrium state. As a consequence, such a system may possess a threshold. Estimations of resting potential, threshold—electrical as well as chemical—and of permeability increase, together with that of the corresponding electrical field strength are shown to have the correct order of magnitude. A possible way to derive the one-factor theory from a physical mechanism as considered here is outlined. It is pointed out that the dependence of the thresholds and the permeability changes on several parameters might be calculated on the basis of such a mechanism. As an example it is shown how in principle some of these relations are derived. Furthermore, the time course of excitatory disturbance for different intensities of the initial disturbance are derived theoretically for the case of chemical stimulation. The curves so obtained show a striking similarity in all the characteristic features with the corresponding ones which are obtained experimentally for the case of electrical stimulation of nerve. These results suggest that response to electrical and chemical stimulation is based on a common threshold phenomenon such as considered here. Finally, a more detailed mathematical description, which takes into account explicitly the diffusion of K+ ions through the membrane for a finite thickness of the membrane is outlined. The equations obtained, which seem to be infeasible of solution at the present time, suggest that it is plausible that relaxation oscillations with a threshold can be derived on the basis of such a mechanism as proposed here. Qualitative agreement with experimental evidence is indicated.

The resonance of the arterial system

The arterial system is assumed to consist of two elastic chambers connected by a conducting channel. It is assumed that a current of fluid enters one chamber, whereas the other chamber is drained by a pipe with a certain peripheral resistance. The continuity of the fluid is described by a differential equation for each chamber. The inertia resistance of the conducting channel is taken into consideration. It is shown that the system may possess a resonance frequency. The latter, if it exists, as well as the damping coefficients are expressed in terms of the elastic moduli of the chambers, the conductivity of the channel, and the peripheral resistance. It is shown that with plausible values of the latter variables the resonance frequency as determined theoretically has the right order of magnitude as found experimentally.

Some types of relaxation oscillations as models of all-or-none phenomena.

After some general remarks the results of the approximation method of Kryloff and Bogoliuboff are applied to show the existence of a threshold for a simple non-linear differential equation of a special form.

On the adsorption of ions at charged sites in an electric field: II

A study is made of the adsorption of one kind of monovalent positive ion at a long chain of alternating monovalent negative fixed charged (“lattice”) and uncharged (“interstitial”) sites both of one type in an electric field. Considering only nearest neighbor interactions an expression is obtained for the grand partition function. The fractions of sites of both types which are occupied and unoccupied are determined. It is shown that an equilibrium constant can be defined for the adsorption of ions at oppositely charged sites.

Studies in quantum biology. II. The mobile electron characteristics of tryptophan+ in relation to those of FMN−, FMNH and FMNH 2 +

By the same method as in the previous paper by the author (Bull. Math. Biophysics,23, 55–68, 1961), the mobile electron system is studied in tryptophan which has donated an electron to another molecule. The obtained net charge distribution of tryptophan+ is compared with those of the semiquinones of riboflavin (FMN−, FMNH and FMNH 2 + ). From this comparison, a suggestion is obtained for the structure of each of two charge transfer complexes between riboflavin and tryptophan in neutral and acid solution.

On spontaneous discharge obtained from a physicochemical model of excitation

It is shown that a slight modification of a model of excitatory phenomena in irritable tissues, which has been treated before, exhibits spontaneous oscillations. The frequency of these oscillations and the time-course of the potential across the model membrane have been determined, together with the dependence of some of their characteristics on some important parameters, particularly (Ca++).

Contributions to the mathematical biophysics of branched circulatory systems

A derivation is given of the reflection coefficient of pressure waves in a vessel whose end branches into many smaller vessles. This coefficient depends on the number of these smaller vessels and their sizes relative to the size of the main vessel. Estimations are made of the order of magnitude of the coefficient. Assuming the main vessel to be of the order of size of an artery, it is shown that the reflection coefficient has a value close to one for reflections at branchings into vessels of arteriolar size. It is pointed out that the result may support the idea that the standing waves in the arterial system are due to reflections at the site of the arterioles.