Contributions to the mathematical biophysics of organic form I. Formation of cavities in cellular aggregates

The action of diffusion forces in aggregates of metabolising cells is studied mathematically, and it is shown that under definite conditions these forces may lead to the formation of inner cavities inside of the cellular aggregate. Different quantitative relations are derived and the possible bearing of these results on some embryological phenomena is discussed. A possible application of these considerations to the theory of formation of vacuoles is also discussed.     

Contributions to the mathematical biophysics of organic form III. Deformation of shell shaped cellular aggregates

The average concentrations of a substance diffusing into or from an open spherical shell, in which it is consumed or produced at a constant rate, are calculated by the approximation method. An application of the result to the problem of deformation of such a shell under the influence of diffusion forces is indicated.     

Contributions to the theory of organic form: The intestinal tract

By making some assumptions concerning the symmetry of certain “classes” of vertebrates and other assumptions concerning the mode of absorption of food by the small intestine, an equation is developed which gives a relation between the length of the small intestine and the total mass of an animal. The equation contains parameters which depend upon the category of vertebrates (carnivorous, omnivorous, etc.) with which the animal is to be associated. The paper also contains a brief discussion of the parameters of the equation and some of the implications which stem from assumptions about their relative magnitudes. In particular, it is noted that the constant of proportionality in the equation which relates intestine length to a power of the body mass is found to be inversely proportional to the square of the “energy content” of the food which makes up the animal's diet. Some suggestions are offered which may lead to an experimental evaluation of the parameters.     

Physio-chemical mathematical theory for transitions in cellular metabolism: An aspect of carcinogenesis

It is pointed out that the non-monotonic character of the chemical reaction rate expressions, together with the relative magnitude of the diffusivity constants, is likely to engender a multiplicity of locally stable steady-state solutions to the system of reaction-diffusion equations for the concentration distributions of molecular species through the volume of a living cell. A transition in cellular metabolism, i.e., the dynamical evolution from an initial locally stable steady-state solution for the concentration distributions to another distinct locally stable steady-state solution, can be induced by an etiologic agent which modifies the rate expressions significantly during an interval of time. Global inequality analysis is employed to derive a condition on the modified rate expressions that is sufficient to guarantee the occurrence of such a transition in cellular metabolism. The possibility of a transition induced by a chemical carcinogen is investigated by applying the latter sufficient condition, and it is found that the statistical frequency of carcinogenesis should depend essentially on the magnitude of the grouping (T ^{2 − α} D ^α) for a total doseD of carcinogen administered to an animal at a uniform rate (D/T) over a time interval of durationT, where α is a certain positive number less than 1. This theoretical result is shown to be supported by the available experimental evidence.     

Contributions to the mathematical theory of epidemics—III. Further studies of the problem of endemicity

(1) The mathematical investigation of the progress of an infectious disease in a community of susceptible individuals has been extended to include the case where members of the community are removed as the result of some general cause of death acting according to constant non-specific death rates, as well as by death from the disease itself. Under the more general conditions here dealt with the main conclusions arrived at in the previous paper remain qualitatively unaltered. The limitations which remain are that the susceptibility and the infective power of the individual are supposed to be independent of his age, and further that specific individual immunity does not exist in the sense that the part of the population which escapes infection is assumed to be just as susceptible as the whole population would have been if it had not been infected. (2) In the general case a unique steady state is found to exist provided that certain relatively simple conditions are satisfied. In the special cases considered a unique steady state in general exists when these conditions continue to be satisfied; but in particular instances, when these conditions are not satisfied, unique steady states will exist provided that certain other requirements are fulfilled. (3) Increase of birth rates, in general, increases both the absolute and the relative prevalence of the disease in its steady state. The effect of increase in the non-specific death rates is less simple, but has been worked out at some length. Decrease in the infectivity of the disease or in the susceptibility of the uninfected results in an increase in the whole population density as well as in an increase in the number of infected. The effect upon the relative incidence of the disease cannot be simply expressed, but it has been worked out in detail in the text. In the absence of immigration, and with the birth rates and also the non-specific death rates equal for virgins and recovered, variation in infectivity or susceptibility will not alter the relative incidence of the disease. The total population, however, will increase with decrease of either of these two factors, whilst the number of diseased will also increase proportionately. (4) Two types of threshold values have been encountered. In the first type the quantity in question must initially exceed the threshold value if the event or process is to occur in the population. Two examples of this type have been found, namely, in Cases (4) and (2'd).(ABSTRACT TRUNCATED AT 400 WORDS)     

Contributions to the mathematical theory of the stability of metabolizing systems with variable surface tension

The paper investigates the stability of a metabolizing system subject to surface tension forces only. In line with some earlier work of N. Rashevsky, the surface tension is considered as a function of the concentration of the metabolite. Different theoretically interesting cases are discussed. The study indicates that surface tension forces alone are not likely to produce phenomena similar to those of cell division.     

Studies in the physicomathematical theory of organic form

In continuation of previous studies the theory is developed on the assumption that the form of any organism is determined by requirements to perform definite biological functions. A previously outlined theory of the form of plants is developed further, showing how the conditions of mechanical strength together with the specifications of the total mass and metabolism, may quantitatively determine not only the general form of the plant, but even the number, size, shape and shades of the leaves. Next the form of animals, as required by mechanical conditions and by the different types of possible locomotions is discussed. A mathematical theory of locomotion of snakes in relation to their shape is outlined. Next is discussed the form and locomotion of quadrupeds. A number of theoretical relations, which describe the shape of an animal, are derived and compared to available observations. After that the theory of flight of birds and insects is discussed, and again some form relations comparable with observations, are discussed. Finally a set of equations is outlined, which determines not only the external shape, but also the internal structure of animals. Different relations pertaining to some inner organs are derived and compared with available observations. The paper ends with a brief discussion on the shape of unicellular organisms.     

Contributions to the mathematical biophysics of visual aesthetics

An expression for the intensity of central excitation corresponding to the perception of an angle is derived. A general expression for the aesthetic value of certain types of polygonal patterns is derived there-from and compared with experimental values obtained by the rank order method. The theory is found to agree with the experiment within a limit of about 10%.     

Contributions to the mathematical biophysics of the cardiovascular system

The reflection of pressure waves in a fluid enclosed within a tube with an elastic wall is studied for the case of a localized change in diameter of the tube. The concept of impedance is introduced. The relation of the reflection characteristics of the parts of the tube at either side of the change is derived on the basis of the continuity of pressure and mass flow at the site of the change. This relations is used to derive the expression for the ratio of the pressure oscillations measured in front of, and behind, the constriction in terms of the constants of the system. As a result, a method is indicated to locate the coarctation from measurements of the pressures in front of, and behind it.     

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.     

Contributions to the mathematical biophysics of the central nervous system

Various neural mechanisms are considered which deal with point to point correspondence between two sets of neural elements with a smaller number of conducting elements between them; the transmission of nerve impulses in a limited range of intensities; movement of the transmission of excitation along a contour; the reaction to the size of an object independent of its distance; and an interpretation of the effect of a warning stimulus and of stimulus intensity upon reaction time. For the latter cases a comparison of the theoretical equations is made with some of the available experimental data, and a general agrement is found.     

Contributions of cytoplasmic factors to in vitro cellular senescence.

Mass populations of normal human lung fibroblasts were enucleated by centrifugation at greater than or equal to 25,000 g in 4 mug/ml cytochalasin B. The 1 per cent of cells that did not enucleate where rendered nonviable by treatment with mitomycin C. Whole cells were poisoned with a 99 per cent lethal dose of the sulfhydryl reagent iodoacetate. The washed cells were then mixed with the anucleate cytoplasms, fused with inactivated Sendai virus, and planted in rotenone for 20 hours. Whereas normal cells are able to survive this rotenone treatment, the 1 per cent surviving iodoacetate-treated cells cannot withstand this additional stress. However, iodoacetate treated cells that fuse to untreated cytoplasms receive sufficient amounts of active enzymes to allow them to survive. Since this selective system does not rely on using enzymatic mutants, it should permit the selection of hybrids between anucleate cytoplasms and any type of whole cell. Cytoplasmic hybrids were cultured in order to determine their proliferative capacity. The life-spans of cytoplasmic hybrids between young and old cells were compared to those of young/young and old/old controls. Cytoplasmic factors do not appear to control in vitro cellular senescence.     

Contributions to the theory of active transport

A system consisting of two solutions separated by a membrane may be in one of four possible states: (1) transient, (2) steady, (3) equilibrium, or (4) pseudo-equilibrium. The latter state denotes that in the solutions the net flow of all components is zero but at least one of the components is not in thermodynamic equilibrium. Transient and steady-state systems may or may not have active transport. Thus only systems in either equilibrium or pseudo-equilibrium are considered in this paper, since the former indicates that there is no active transport, whereas in the latter case there always is active transport. This simplifies the problem of finding whether a system does or does not have an active transport mechanism, since it is frequently fairly easy to determine experimentally whether a system is in equilibrium or pseudo-equilibrium. The assumption that electric neutrality exists within very thin membranes is shown not to be valid. However, electric neutrality does exist in the solutions in a system in a pseudo-equilibrium state with fixed charges and impermeative ions. It is then shown how the presence and sign of an electric potential may be found by use of electroneutrality. The mechanism of active transport may be due to a general force acting on all particles of a particular component or to an individual force acting on the individual particles of a particular component. A general solvent flow or a diffusion drag force illustrates the first mechanism while the second is accomplished by either a carrier or a Maxwell Demon. The general type of active transport has been extensively treated in the literature, while the individual type has not been treated in a generalized form. Therefore, the individual type of active transport is discussed at length, and a simple illustrative model is intensively analyzed. Following this, there is a discossion of the Maxwell Demon and some models of it are presented. This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under contract No. AF 18(600) 1454.     

Contributions to the theory of imitative behavior

The imitation effects in a social group depend both on the size of the group and on the distribution of a certain psychobiological quantity ϕ which measures the tendency of an individual towards a given behavior. The distribution function of ϕ determines the ratio μ of the individuals in the society who adopt a given behavior. When the size of the social group is not too large, the actual distribution of ϕ will deviate from the most probable one, and therefore communities of the same size and having the same parameters may have different values of μ. Approximate equations are developed which give the probability of a given μ for a group of a given size. Possible effects of interactions of communities of different sizes are briefly discussed. A generalization of the theory of imitative behavior to any number of mutually exclusive behaviors is given, and its possible sociological implications are discussed.     

II. Contributions to the Ornithology of the East African Islands.

The sea-bird breeding stations in East African waters are enumerated. Between the Lamu archipelago, where many species nest, and the Rovuma the most important station is Latham Island with its Booby colony ( S. dactylatra melanops ), which existed at least 115 years ago. Less than half a dozen terneries (mainly S. dougallii and S. fuscata) are known, and no gullery. Reef-Herons, the only sea-birds to breed on bush-covered islets or in mangroves, are more widespread than the other species. The Terns lay about July, the Reef-Herons a month earlier, and the Gannets about four months later in most years.
Early records of birds in Zanzibar and Pemba are discussed specifically. The recent extinction of Francolins in Zanzibar is taken as proved.
The land-birds of Mafia and the neighbouring islets are listed and discussed, mainly on the basis of a recent collection. The avifauna is purely East African, devoid of endemics, and, considering the size of Mafia and its proximity to the mainland, astonishingly depauperate in some respects. Thirteen families (including Psittacidæ, Coliidæ, Picidæ, Alaudidæ, Timaliidæ, and Zosteropidæ) and several other coastal species are not represented. The phenology is discussed.     

Contributions of genomics to life-history theory

Life-history theory seeks to understand the factors that produce variation in life histories that are found both among and within species. At the organismal level there is a well developed mathematical framework, and an important focus of the current research is determining the biological underpinnings of this framework, with particular attention to the causal mechanisms that underlie trade-offs. Genomic approaches are proving useful in addressing this issue.