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.     

Contributions to the mathematical theory of organic form. II Asymmetric metabolism of cellular aggregates

The case is studied of a cellular aggregate forming a hollow spherical shell, the physical constants and the metabolic rate being non-uniform throughout the shell. This leads to asymmetries of concentration distributions, which are calculated here for some simple cases.     

A physicomathematical model of sodium balance

A model is introduced in which the reabsorption of sodium is governed by an enzymatic process. This process is in turn assumed to be influenced by the extracellular volume which depends on the amount of sodium in the body at a given time. The model allows for damped oscillations when the sodium intake lies within range of values and thus can account for observed oscillations. This investigation was supported by the U.S. Public Health Service (grants 5-K6-GM-18, 420 and A-4668), and by a grant from the Dr. Wallace C. and Clara A. Abbott Memorial Fund of the University of Chicago.     

The theoretical physics of the cell as a basis for a general physico-chemical theory of organic form

Summary It is shown by thermodynamical considerations, that in general regardless to the particular nature of a cell, there must be two kinds of forces between cells: a force of repulsion, due to metabolism, and a force of attraction, due to irritability. It is shown, that the various geometrical forms, which cellular aggregates may assume under the influence of those forces, correspond in general features to various forms, found in nature. In conclusion I wish to express my profound indebtedness to ProfessorDavenport Hooker, for frequent discussions of the subject and a great amount of valuable information. With 1 Text-figure     

Studies in the mathematical theory of excitation

The general linear two-factor nerve-excitation theory of the type of Rashevsky and Hill is discussed and normal forms are derived. It is shown that in some cases these equations are not reducible to the Rashevsky form. Most notable is the case in which the solutions are damped periodic functions. It is shown that in this case one or more—in some cases infinitely many—discharges are predictable, following the application of a constant stimulusS. The number of discharges increases withS, but the frequency is a constant, characteristic of the fiber and independent ofS.     

Microbial production of biogas form organic wastes

Biogas production through bacterial digestion of water hyacinth, rice straw and sawdust inoculated with cow dung was investigated in a batch loading of laboratory digestors. It was found that the overall gas yield was 368.00 l from water hyacinth having 77.13% methane, 310.00 l from rice straw with 77.80% methane, and 144.80 l from sandust containing 62.28% methane. Along with the production of gas, the change in pH and microbial population were also monitored. The pH of the digestion mixture was initially low and found to be increasing as the digestion progressed. The total microbial population of bacteria, fungi, actinomycetes, cellulolytic organisms, and coliform bacteria were found to decrease with time.     

Asymptotic birth trajectories in the discrete form of stable population theory

This paper seeks to supply a hitherto missing link in the reconciliation of the discrete and continuous formulations of stable population theory—see, for example, Keyfitz (“Introduction to the Mathematics of Population,” Chap. 8, Addison-Wesley, Reading, Mass., 1968). We show that the real exponential component of the birth trajectory has a non-zero (in fact positive) coefficient. Consequently, there is no possibility that oscillatory terms in the series expression for the birth trajectory can be dominant. This is shown by obtaining a formula for the desired coefficient.     

Niche theory and plant growth form

Plant growth form diversity (GFD) is high in the vegetation of North American deserts, and increases from north (Great Basin Desert) to south (Sonoran Desert). While abiotic features (annual temperature, precipitation, and seasonality) appear to limit the range of desert plant GFD, biotic features associated with the coexisting plants at a site, and their GF distribution, add further constraints. Climate may constrain the GF options at certain sites and select for some degree of GF convergence there, but within sites other species in the vegetation select for GF segregation that fosters the local coexistence of species. In this paper GF variation is viewed along structural niche axes, and related to classical niche theory; several corollaries of the theory are examined in the light of plant GF patterns. These are: a) regular spacing of species on the structural niche axis, and the concept of limiting similarity; b) niche axis complementarity, such that species dissimilar in position on one axis, e.g. GF, are similar in position on other axes, e.g. habitat or substrate, and vice versa; c) niche shifts in GF within species are expected, and occur, as the suite of coexisting species varies among sites with similar climate; d) in some desert plant guilds species with very similar GF do not coexist at a site, but act as geographical replacements in different sites.     

Physicomathematical aspects of some problems of organic form

In connection with previous mathematical studies on cell polarity, the possible application of the results obtained before to different embryological phenomena is discussed. Methods for a quantitative mathematical approach to such phenomena as gastrulation, formation of different folds, closing of a half blastula, etc. are outlined.     

Studies in the stability theory of complex homogeneous kinetics

Summary The results of an investigation of the formal stability behavior of equivalent classes of asymptotically stable kinetic equations (generated by well-posed stability constraint specifications), general enough so as to include non-polynomial kinetics, are presented in detail. The qualitative properties of such equivalent classes — especially with respect to physical admissibility and fluctuation regression, including bounds on generalized relaxation times—are deduced without resorting to an explicit solution. The immediate motivation for the study arose from the need for a comprehensive theory when steric interactions in biosynthetic networks are explicitly taken into account. It is known that such interactions result in non-polynomial kinetics.     

Studies in the stability theory of complex homogeneous kinetics

Summary Fundamental decomposition of the equation of chemical flux of the exhaustive equivalent classes of open, as well as closed, kinetic systems asymptotically stable with respect to a unique steady-state, gives rise to two flux components of which one can be shown to be strictly decrescent, the other being strictly conservative, with respect to the scalar potential V(x) of the stability constraint specification generating the equivalent classes. In this paper, which constitutes the second part of a series, the purely formal, as well as the biological consequences of the fundamental decomposition are considered.     

The theory of tree bole and branch form

Summary Working from the general postulate that natural selection of plant form operates so as to maximize the survival potential of a species, this paper examines the hypothesis that the mechanical support of tree foliage must approach optimality in the use of wood, i.e., that tree stems and branches will have optimal form with respect to the amount of support tissue. Mathematical models of bole and branch form are presented, based on the proposition that either wind or gravity are the primary limiting factors for tree size and shape. Predictions of trunk and branch diameter as a function of tree size were tested with dimensional measurements ofPopulus tremuloides. The individual stems were selected from close-grown stands of differing ages. For small and intermediate trees, trunk diameter is such that stems have only 1.6 times as much wood as the minimum required to keep the tree from buckling under its own weight due to elastic instability. Branch diameters are shown to be close to the minimum required to maintain the spatial position of growing branches, as well as withstand wind forces. This minimal branch cost not only reduces the load which the stem must support against elastic instability, but allows the crown to flex in high winds. The flexing, in turn, reduces the drag force exerted by the wind on the trunk. Thus, the hypothesis that the observed tree form is an optimal design cannot be rejected on the basis of these results. Additional studies are planned with respect to optimal foliage distribution.     

Homology and a generative theory of biological form

Homology continues to be a concept of central importance in the study of phylogenetic relations, but its relation to ontogenetic processes remains problematical. A definition of homology in terms of equivalent morphogenetic processes is defined and applied to the comparative study of tetrapod limbs. This allows for a consistent treatment of relations of similarity and difference of appendage structure in vertebrates, and the distinction between fishes fins and tetrapod limbs in terms of the concept of equivalence is described. The role of genes can also be clarified in this context, in particular the influence of the Hox 4 complex in determining digit character and the homeotic transformations that arise from changes in their expression patterns. It is argued that these observations are not compatible with the notion of homology between individual digits (I, II, III, etc.) across the tetrapods, and that homology cannot be consistently identified with gene action. The relations between homology and the properties of the morphogenetic limb field are discussed.     

Meckel the Younger and his epistemology of organic form: Morphology in the pre-Gegenbaurian age

Summary The goal of our paper is to investigate Meckel’s epistemology of organic form, based on study of his original publications. Johann Friedrich Meckel the Younger (1781–1833) was one of the leading figures of German morphology in the early 19th century. Historiographic studies on morphology in this time period show, that biological research was largely preoccupied with questions about the relationship between form and function. Investigations into Meckel’s epistemology of organic form can contribute to our understanding of the development of morphology in the pre-Gegenbaurian age.