Dimensional analysis in mathematical biology I. General discussion

Dimensional analysis is discussed from the viewpoint of its basic group properties and shown to be an algebraic Abelian group that is useful for analysis of physical measurements. The application of the method to various types of equations and the formulation of previously unclassified dimensions are discussed. Functional dimensional analysis is applied to the problems of cell size and biomass proliferation; future applications are also noted. A number of dimensionless terms have been formulated for cellular physiochemical phenomena. They apparently represent the first systematic study of biological dimensionless numbers recorded in the literature. A dimensionless proliferation law is suggested. A brief analysis of the physical dimensionality associated with information measures is carried out. Entropy and “information” are shown to be completely different in their dimensional meaning; other informational measures of possible interest in biology are proposed. The dimensional coding and computor analysis of biomathematical equations is suggested.     

Theory of biological similarities,nondimensional parameters and invariant numbers

In previous papers (1955–1957) a theory of biological similarity was established, assuming that the limits are the mechanical and the electrodynamical similarity criteria. The range of this theory lies between the coefficient of the time exponent (γ) for mechanical (0.5γ) and electrodynamical (1.0γ) similarities, being the mode 0.93γ. Moreover, for certain functions this restricted theoretical range should be extended to the hydrodynamical similarity criterion (2γ), so that the dimensionless numbers commonly used in Physics (Reynolds, Froude, Weber, etc.) can be included within the total range (0.5–2γ) of biological similarities. From dimensional analysis of physiological, functions it was possible to obtain, by means of dimensional and solution matrices, a group of “nondimensional numbers” by applying Buckingham's Pi-theorem. Nevertheless, only if a single similarity criterion was applied, the residual weight exponent was exactly zero; in all other instances the weight exponent was not zero, due to the existence of a range for biological similarities and to the statistical meaning of exponent (b) of the allometric equations. From the similarity criteria “invariant numbers” can be obtained, by means of which it is possible to establish correlations between numerous morphological and physiological characteristics of a particular system (circulation, respiration, metabolism, etc.).     

“Force-Talk” in Evolutionary Explanation: Metaphors and Misconceptions

The notion of “pressure” as an evolutionary “force” that “causes” evolution is a pervasive linguistic feature of biology textbooks, journal articles, and student explanatory discourse. We investigated the consequences of using a textbook and curriculum that incorporate so-called force-talk. We examined the frequency with which biology majors spontaneously used notions of evolutionary “pressures” in their explanations, students’ definitions and explanations of what they meant when they used pressures, and the structure of explanatory models that incorporated evolutionary pressures and forces. We found that 12–20 percent of undergraduates spontaneously used “pressures” and/or “forces” as explanatory factors but significantly more often in trait gain scenarios than in trait loss scenarios. The majority of explanations using “force-talk” were characterized by faulty evolutionary reasoning. We discuss the conceptual similarity between faulty notions of evolutionary pressures and linguists’ force-dynamic models of everyday reasoning and ultimately question the appropriateness of force-talk in evolution education.     

Age changes of the carpal bones in macaques: Application of the fourier analysis on the radiographs

A method based on the Fourier analysis is proposed, which describes and analyzes the contour morphology of carpal bones by separating morphology into factors of shape and size. Here, “size” refers to the average diameter of the contour. The “shape” is expressed byshape factors which are derived from the Fourier series and the “shape” of a monkey is expressed by ashape index which is calculated fromshape factors. The age change in the morphology of the lunate and capitate ofMacaca fuscata fuscata was analyzed by this method. The development of “shape” approximately completes by 3 years of age, whereas increase in “size” begins its spurt at that age as do body weight and anterior trunk length. By applying this method to other macaque species, it was found thatM. mulatta, M. f. yakui andM. cyclopis exhibit similar patterns of growth and development of carpal bones to those ofM. f. fuscata. Patterns found inM. fascicularis differ in that its bones develop faster than in the other macaques with respect to the “shape,” but remain small with respect to the “size.”     

Don’t Call it “Darwinism”

Evolutionary biology owes much to Charles Darwin, whose discussions of common descent and natural selection provide the foundations of the discipline. But evolutionary biology has expanded well beyond its foundations to encompass many theories and concepts unknown in the 19th century. The term “Darwinism” is, therefore, ambiguous and misleading. Compounding the problem of “Darwinism” is the hijacking of the term by creationists to portray evolution as a dangerous ideology—an “ism”—that has no place in the science classroom. When scientists and teachers use “Darwinism” as synonymous with evolutionary biology, it reinforces such a misleading portrayal and hinders efforts to present the scientific standing of evolution accurately. Accordingly, the term “Darwinism” should be abandoned as a synonym for evolutionary biology.     

Some generalized conjugacy theorems and the concepts of fitness and survival in logistic growth models

Current research into the dynamics of iterative ecological and biological models has lead to a number of theorems concerning the existence of various types of iterative dynamical behavior. In particular, much study has been done on the dynamical behavior of the “simplest dynamical system”f _b(x)=bx(1−x), which is just the canonical discrete form of logistic growth equations found in ecology, sociobiology, and population biology. In this paper, we make use of some of the techniques and concepts of topological dynamics to construct a number of generalized conjugacy theorems. These theorems are then used to demonstrate that the mappingf _b has a number of conjugacy classes in which the dynamics of the iterates is equivalent to within a change of variables. The concepts of fitness and survival in logistic equations are then shown to be independent, if we follow certain intuitive definitions for these concepts. This conclusion follows from a comparison of the conjugacy classes of the functionf _b and the extinction sets off _b.     

The biology behind lichenometric dating curves

Lichenometry is used to date late-Holocene terminal moraines that record glacier fluctuations. Traditionally, it relies upon dating curves that relate diameters of the largest lichens in a population to surface ages. Although widely used, the technique remains controversial, in part because lichen biology is poorly understood. We use size-frequency distributions of lichens growing on well-dated surfaces to fit demographic models for Rhizocarpon geographicum and Pseudophebe pubescens, two species commonly used for lichenometry. We show that both species suffer from substantial mortality of 2–3% per year, and grow slowest when young-trends that explain a long-standing contradiction between the literatures of lichenometry and lichen biology. Lichenometrists interpret the shape of typical dating curves to indicate a period of rapid juvenile “great growth,” contrary to the growth patterns expected by biologists. With a simulation, we show how the “great growth” pattern can be explained by mortality alone, which ensures that early colonists are rarely found on the oldest surfaces. The consistency of our model predictions with biological theory and observations, and with dozens of lichenometric calibration curves from around the world, suggests opportunities to assess quantitatively the accuracy and utility of this common dating technique.     

On the Structural Regularity in Nucleobases and Amino Acids and Relationship to the Origin and Evolution of the Genetic Code

To explore how chemical structures of both nucleobases and amino acids may have played a role in shaping the genetic code, numbers of sp² hybrid nitrogen atoms in nucleobases were taken as a determinative measure for empirical stereo-electronic property to analyze the genetic code. Results revealed that amino acid hydropathy correlates strongly with the sp² nitrogen atom numbers in nucleobases rather than with the overall electronic property such as redox potentials of the bases, reflecting that stereo-electronic property of bases may play a role. In the rearranged code, five simple but stereo-structurally distinctive amino acids (Gly, Pro, Val, Thr and Ala) and their codon quartets form a crossed intersection “core”. Secondly, a re-categorization of the amino acids according to their β-carbon stereochemistry, verified by charge density (at β-carbon) calculation, results in five groups of stereo-structurally distinctive amino acids, the group leaders of which are Gly, Pro, Val, Thr and Ala, remarkably overlapping the above “core”. These two lines of independent observations provide empirical arguments for a contention that a seemingly “frozen” “core” could have formed at a certain evolutionary stage. The possible existence of this codon “core” is in conformity with a previous evolutionary model whereby stereochemical interactions may have shaped the code. Moreover, the genetic code listed in UCGA succession together with this codon “core” has recently facilitated an identification of the unprecedented icosikaioctagon symmetry and bi-pyramidal nature of the genetic code.     

Comparative Epistemology: Contours of a Research Program

This article addresses the question whether and how literary documents can be used to further our understanding of a number of key issues on the agenda of the philosophy of biology such as “complexity” and “reductionism”. Kant already granted a certain respectability to aesthetical experiences of nature in his third Critique. Subsequently, the philosophical movement known as phenomenology often used literary sources and literary techniques to criticize and question mainstream laboratory science. The article discusses a number of literary documents, from Moby-Dick to Jurassic Park, that explicitly stage a confrontation between scientific and non-scientific ways of experiencing and understanding the natural world. Special attention is given to the work of Michael Crichton. Its relevance for the philosophy of biology is pointed out.     

Distinguished words in data sequences: Analysis and applications to neural coding and other fields

This paper concerns sequences of letters in which certain “distinguished” words are of interest. Such sequences arise as data in numerous fields including genetics and neuroscience. A probability distribution is given for the number of occurrences of a chosen word in a randomized sequence of letters. Such words are considered “favored” if they occur more than expected at random. Favored words have been discovered in nerve impulse trains and may reflect a neural coding scheme. This article is dedicated to my mother, Margaret Oakley Dayhoff, whose enthusiasm encouraged me to pursue research in mathematical biology.     

Evolutionary ecology of clonal-bisexual complexes in spined loaches from genus <Emphasis Type="Italic">Cobitis</Emphasis> (Pisces,Cobitidae)

The objective of this work was to analyze several aspects of evolutionary ecology of the spined loach clonal-bisexual complexes. This analysis demonstrates that the twofold population growth rate in clonal forms compared with bisexual species is not a proper feature of spined loaches, because their clonal forms are polyploids with the larger eggs and correspondingly the lower individual fecundity. Both the larger nutrient stores in eggs and the larger hatchlings provide certain ecological advantages for clonal spined loach triploids at early stages of their ontogeny and possibly ensure the higher fitness in a local environment. The clonal-bisexual spined loach complexes have reduced structure usually including one bisexual species and the only clonal form. This reduction may be caused by several factors: 1) the competition between different forms and subsequent superseding of some of them; 2) a particular manner of colonization of new areas by different bisexual and clonal forms; 3) the absorption of one of parental species for hybrid origins of clonal forms (hypothesis of absorption). The relative numbers of bisexual and unisexual forms in different clonal-bisexual complexes depend on “family ties” between clonal forms and “host” bisexual species; their ratios are subjected to temporal dynamics caused by both external (environmental changes) and internal (automatic fluctuating) factors.     

J D Bernal (1901–1971) in perspective

Conclusion We must conclude that the sub-title of Bernal’s “The Social Function of Science” — “What science does: what science could do” is still the relevant challenge and indicates Bernal’s chief contribution, besides the foundation of molecular biology to our civilization. It is manifest that resources spent on armaments are a monstrous pathological symptom of our social structure. The ancient problem of “what is property” and what may be “owned” and by whom or by what organs of society is awakening.     

Hematopoietic cell renewal systems: mechanisms of coping and failing after chronic exposure to ionizing radiation

On the occasion of the first international workshop on systems radiation biology we review the role of cell renewal systems in maintaining the integrity of the mammalian organism after irradiation. First, 11 radiation emergencies characterized by chronic or protracted exposure of the human beings to ionizing irradiation were “revisited”. The data provide evidence to suggest that at a daily exposure of about 10–100 mSv, humans are capable of coping with the excess cell loss for weeks or even many months without hematopoietic organ failure. Below 10 mSv/day, the organisms show some cellular or subcellular indicators of response. At dose rates above 100 mSv/day, a progressive shortening of the life span of the irradiated organism is observed. To elucidate the mechanisms relevant to tolerance or failure, the Megakaryocyte–thrombocyte cell renewal system was investigated. A biomathematical model of this system was developed to simulate the development of thrombocyte concentration as a function of time after onset of chronic radiation exposure. The hematological data were taken from experimental chronic irradiation studies with dogs at the Argonne National Laboratory, USA. The results of thrombocyte response patterns are compatible with the notion of an “excess cell loss” (compared to the steady-state) in all proliferative cell compartments, including the stem cell pool. The “excess cell loss” is a function of the daily irradiation dose rate. Once the stem cell pool is approaching an exhaustion level, a “turbulence region” is reached. Then it takes a very little additional stress for the system to fail. We conclude that in mammalian radiation biology (including radiation medicine), it is important to understand the physiology and pathophysiology of cell renewal systems in order to allow predicting the development of radiation induced lesions.     

Some remarks on the boolean algebra of nervous nets in mathematical biophysics

Recent demonstration by the author has shown that the fundamental equations of the mathematical biophysics of the central nervous system can be considered as describing the behavior of very large numbers of neurons, of which each one follows discontinuous laws, such as discussed by W. S. McCulloch and W. Pitts. In that light some of the old problems are discussed. The comparative merits of the “microscopic” and “macroscopic” approaches are discussed for the problem of the point to point correspondence between the retina and the cortex, with the number of connecting fibers much less than the number of cells. Some aspects of discrimination of intensities are also discussed. Finally, a few generalizations of the McCulloch-Pitts treatment are suggested, and a nervous network is constructed which illustrates some aspects of the perception of numbers.     

Dimensional approaches to designing better experimental ecosystems: a practitioners guide with examples

Enclosed, experimental ecosystems (“mesocosms”) are now widely used research tools in ecology. However, the small size, short duration and often simplified biological and physical complexity of mesocosm experiments raises questions about extrapolating results from these miniaturized ecosystems to nature. Dimensional analysis, a technique widely used in engineering to create scale models, employs “compensatory distortion” as a means of maintaining functional similarity in properties and relationships of interest. An earlier paper outlined a general approach to applying dimensional analysis to the construction and interpretation of mesocosm experiments (Petersen and Hastings in Am Nat 157:324, 2001). In this paper we use examples, largely drawn from the aquatic literature, to illustrate how dimensional approaches might be used to maintain key ecological properties. Such key properties include effective habitat size, environmental variability, vertical and horizontal gradients, and interactions among habitats. We distinguish both continuous and discrete approaches that can be used to achieve functional similarity through compensatory distortion. In addition to its potential as a tool for improving the realism of experimental ecosystems, the dimensional approach points towards new options for developing, testing and advancing our understanding of scaling relationships in nature. Electronic Supplementary Material Supplementary material is available for this article at     

In vitro separation of a rose chimera

“Fairmount 1 thorny” (“FM1 thorny”) (a Rosa multiflora Thunb ex. J. Murr.) and a thornless sport of “FM1 thorny” (“Fairmount 1” (“FM1”)) were established in vitro to investigate chimeral segregation under various levels of BA and to obtain a pure thornless rose. While the chimeral thornless sport was expected to segregate in vitro and yield both thorny and thornless plantlets, “FM1 thorny” was to yield only thorny plants. “FM1” segregated in vitro into its constituent genotypes and yielded thorny and thornless plantlets, suggesting that “FM1” is chimeral. “FM1 thorny” produced only thorny plants in vitro. These results indicate that the “FM1 thorny” clone was not chimeral (pure thorny) and that the thornless regenerates of “FM1” did not develop via somaclonal variation. There was a significant linear relationship between increasing BA concentration and the percentage of thorny plants. Among a population of 690 tissue culture derived plants from all the BA experiments, 6 plants were classified as pure thornless plants 1 year later.     

Redefining the X Axis: “Professionals,” “Amateurs” and the Making of Mid-Victorian Biology – A Progress Report

A summary of revisionist accounts of the contextual meaning of`“professional” and “amateur,” as applied to the mid-Victorian X Club, is followed by an analysis of the liberal goals and inner tensions of this coalition of gentlemen specialists and government teachers. The changing status of amateurs is appraised, as are the new sites for the emerging laboratory discipline of “biology.” Various historiographical strategies for recovering the women’s role are considered. The relationship of science journalism to professionalization, and the constructive engagement of X Club publicists with their empowering audiences, are discussed. Finally, the article assesses how far the content and boundary closure of ``biology,' forged by Thomas Henry Huxley, were related to `professional' and political goals. Purebiology’s social and medical roots are examined, and the way inter-professional and wider Darwinian conflicts resulted in anew lexicon of words for the X Clubbers around 1870, including“evolution” and “agnosticism,” as well as “biology.” Biology’srole in the forging of British national identity is discussed, as are its relationship to the social strategies of liberal, Dissenting, and industrial groups in the country, whose authority sustained the new laboratory rhetoric. This revised version was published online in July 2006 with corrections to the Cover Date.     

Generalization and categorization of spectral colors in goldfish. II. Experiments with two and six training wavelengths

In part I of this study (Kitschmann and Neumeyer 2005), goldfish categorized spectral colors only in the sense that wavelengths in a range of about twice as large as the just noticeable difference were treated as similar to a given training wavelength. Now, we trained goldfish on more than one wavelength to prevent very accurate learning. In one experiment goldfish were trained on six adjacent wavelengths with equal numbers of rewards, and, thus, equal numbers of learning events. Generalization tests showed that some wavelengths were chosen more often than others. This indicated that certain spectral ranges are either more attractive or more easily remembered than others. As this is a characteristic of the “focal” colors or centers of color categories in human color vision, we interpret the findings in goldfish accordingly. We conclude (Figs. 5 and 6) that there are four categories in spectral ranges approximately coinciding with the maximal sensitivities of the four cone types, and three categories in-between. Experiments with two training colors indicate that there is no direct transition between categories analogous to human “green” and “red”, but that there is a color analogous to human “yellow” in-between (Figs. 2, 3; Table 1).     

Intrinsic time scaling in survival analysis: Application to biological populations

A method of dimensionless time-scaling based on extrinsic expectation of life at birth but intrinsic to a system generating a survival distribution is introduced. Such scaling allows the survival fraction function and its associated mortality function to serve as Green's functions for their generalized equivalents. i.e. a “population” function and a “death” function. The analytical mechanics of utilizing these concepts are formulated, applied to the classical Gompertz and Weibull survival models, and discussed with respect to biological relevance.     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997