Note on the fusion frequency for stimuli of alternating duration

The case of alternating stimulus patterns of moderately high intensity is considered in terms of the model previously discussed (Landahl, 1957,Bull. Math. Biophysics,19, 157–62). If both of the alternating patterns have the same light-dark ratio, then the relation between the period of the longer and the period of the shorter pattern at the critical flicker frequency is independent of the light-dark ratio and is given by a dimensionless expression which is roughly in agreement with data in which the light-dark ratio is one (C. R. Brown and D. M. Forsyth, 1959,Science,129, 390–91). This research was supported in whole or in part by the U. S. Air Force under Contract AF 49(638)-414 monitored by the Air Force Office of Scientific Research.     

An expression for the rate of return of an egg after artificial deformation

The forces which may be involved in the restoration of a deformed cell to its normal shape are considered. Estimates of the order of magnitude of the forces suggest that the most important forces are those due to surface tension, membrane elasticity and viscosity. An approximate expression is then derived for the rate of return of an elongated or compressed egg. The former expression is compared with data on eggs ofArbacia by E. N. Harvey and H. Shapiro, and it is found to agree sufficiently well with the data. The initial surface force is too low compared with that given by K. S. Cole but various factors are discussed which could contribute to this discrepancy. The value for the net viscosity of the egg is about twice the value arrived at by L. V. Heilbrunn for the egg protoplasm.     

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⁺⁺).     

A note on the mathematical biophysics of central excitation and inhibition

Some relations between the temporally macroscopic theory of central excitation and inhibition and the temporally microscopic theory of nervous nets are suggested.     

Note on the effects of maternal selection in biological populations

The dynamics of the genetic structure of populations under various selection factors are considered. The effect of selection by way of the maternal genotype through fecundity, favorable placement of eggs and such factors as are dependent on the character of the maternal parent is emphasized. This factor can lead to a stable equilibrium with genotype frequencies which would appear to be unstable. A similar result can be obtained as a result of differences in mating probabilities which may arise due to differences in maturation rates. A number of cases are treated to illustrate the dynamics as well as conditions for stable equilibrium.     

Theoretical considerations on potentiation in drug interaction

To account for some of the more important aspects of drug interaction we shall consider a model which can also account for certain general properties of the action of a single drug. A simple model in which there may be enzymatic detoxification of a drug is studied theoretically. The relation between time for appearance of an effect due to the drug and the size of the dose is found to contain the same parameters as the relation between the effectiveness of paired doses and the interval of time between doses. A similar situation holds when the drug is given at a constant rate. When two drugs are administered together, their effect will depend on the manner of interaction, how much of each drug is given, which is given first, and on the interval of time between each administration. A number of plausible types of interaction is considered theoretically in terms of the model, analytical expressions being given for a number of cases. The interaction may be synergistic or antagonistic. In the former case the potentiation may be more than or less than additive depending on the order of delivery and on the time between injections. Methods for the estimation of the parameters from data are discussed.     

On the velocity of conduction in nerve fibers with saltatory transmission

Using an electrical model to represent certain features of a nerve fiber together with a one-factor theory of excitation, an expression is obtained for the velocity of propagation of a nerve impulse along a nerve fiber exhibiting saltatory transmission. The velocity shows a maximum with respect to internodal length. A critical internodal length, a critical radius, and a critical value for the resistance of the external medium are required in order to make transmission of the impulse possible.     

Mathematical biophysics of cell respiration II

The recent approximate method developed by N. Rashevsky in his bookMathematical Biophysics is shown to lead to results equivalent to those obtained in the preceding paper on cell respiration. An estimate is made of the value of the total diffusion resistance to lactic acid for unfertilizedArbacia eggs. Several more general reactions are treated to illustrate the effect of intermediate products. Expressions for the glycolytic coefficient are derived in terms of the parameters of the cell.     

Particle removal by the respiratory system note on the removal of airborne particulates by the human respiratory tract with particular reference to the role of diffusion

Calculations of the contribution of diffusion to lung retention given previously (W. Findeisen,Pflüg. Arch. f. d. ges. Physiol.,236, 367–379, 1935; H. D. Landahl,Bull. Math. Biophysics,12, 43–56, 1950) contained an error appreciably underestimating the role of diffusion for particles less than a few microns in diameter. In the calculations (Landahl, 1950), a schematic representation of the lungs was modified to conform to the actual resting lung volume by increasing the number of alveolar sacs. A recalculation of total deposition with this chematic model gives too large a retention for very small particles. Hence it seemed appropriate to recalculate the retention for various paricle sizes in various regions of the respiratory tract for several respiratory paterns, using the original schematic model but assuming that the estimates reflect conditions in a deflated lung. For a given tidal air the sizes of the alveolar ducts and sacs were assumed to expand so that the total volume of the model was equal to 2.5 liters plus one half of the tidal air, i.e., the mean lung volume. The results of these calculations are appreciably different from those given previously, but the differences are serious only for particles of about 0.1 micron or less. Also given is an approximation method for calculating the probability of impaction of droplets against nasal hairs.     

A note on population growth under random dispersal

An approximation method using a sine function is used to solve the second degree growth equation for the case in which an organism may simultaneously become dispersed throughout a uniform region. The resulting expression for a special case is compared with the expression obtained by R. Barakat (1959,Bull. Math. Biophysics,21, 141–51), giving the first two terms, by an iterative, procedure. The agreement is satisfactory.