Molecular set theory: II. An aspect of the biomathematical theory of sets

In an earlier paper (Molecular Set Theory: I.Bull. Math. Biophysics,22, 285–307, 1960) the author proposed a “Molecular Set Theory” as a formal mathematical meta-theoretic system for representing complex reactions not only of biological interest, but also of general chemical interest. The present paper is a refinement and extension of the earlier work along more formal algebraic lines. For example the beginnings of an algebra of molecular transformations is presented. It also emphasizes that this development, together with the genetical set theory of Woodger's and Rashevsky's set-theoretic contributions to Relational Biology, points to the existence of a biomathematical theory of sets which is not deducible from the general mathematical, abstract theory of sets.     

The mathematical biophysics of Nicolas Rashevsky

N. Rashevsky (1899-1972) was one of the pioneers in the application of mathematics to biology. With the slogan: mathematical biophysics : biology :: mathematical physics ; physics, he proposed the creation of a quantitative theoretical biology. Here, we will give a brief biography, and consider Rashevsky's contributions to mathematical biology including neural nets and relational biology. We conclude that Rashevsky was an important figure in the introduction of quantitative models and methods into biology.     

Topological biology: A note on Rashevsky's transformation T.

In the bio-topological transformation between graphs denoted by (T ⁽¹⁾ X) N. Rashevsky (Bull. Math. Biophysics,18, 173–88, 1956) considers the number of fundamental sets which (a) have only one specialized point as source (and no other sources), (b) have no points in common (are “disjoined”); he proves that this number is an invariant of the transformation. In this note we show that Rashevsky's Theorem can be extended as follows:The number of fundamental sets of the first category is an invariant of the transformation. We must, however, count the subsidiary points of the transformed graph as specialized points. We recall that fundamental sets of the first category are those whose sources consist of specialized points only (Trucco,Bull. Math. Biophysics,18, 65–85, 1956). But in this modified version of the Theorem the fundamental sets may have more than one source and need not be disjoined.     

Mutant Bacteriophages,Frank Macfarlane Burnet,and the Changing Nature of “Genespeak” in the 1930s

In 1936, Frank Macfarlane Burnet published a paper entitled “Induced lysogenicity and the mutation of bacteriophage within lysogenic bacteria,” in which he demonstrated that the introduction of a specific bacteriophage into a bacterial strain consistently and repeatedly imparted a specific property – namely the resistance to a different phage – to the bacterial strain that was originally susceptible to lysis by that second phage. Burnet’s explanation for this change was that the first phage was causing a mutation in the bacterium which rendered it and its successive generations of offspring resistant to lysogenicity. At the time, this idea was a novel one that needed compelling evidence to be accepted. While it is difficult for us today to conceive of mutations and genes outside the context of DNA as the physico-chemical basis of genes, in the mid 1930s, when this paper was published, DNA’s role as the carrier of hereditary information had not yet been discovered and genes and mutations were yet to acquire physical and chemical forms. Also, during that time genes were considered to exist only in organisms capable of sexual modes of replication and the status of bacteria and viruses as organisms capable of containing genes and manifesting mutations was still in question. Burnet’s paper counts among those pieces of work that helped dispel the notion that genes, inheritance and mutations were tied to an organism’s sexual status. In this paper, I analyze the implications of Burnet’s paper for the understanding of various concepts – such as “mutation,” and “gene,” – at the time it was published, and how those understandings shaped the development of the meanings of these terms and our modern conceptions thereof.     

Foliage consumption and larval development of parasitized and unparasitized soybean looper,Pseudoplusia includens [Lep.: Noctuidae], reared on a resistant soybean genotype and effects on an associated parasitoid,Compidosoma truncatellum [Hym.: Encyrtidae]

Field grown foliage from the resistant soybean [Glycine max (L.) Merrill] breeding line GAT “81–327” and the susceptible cultivar “Ransom” was used to rear unparasitized larvae of the soybean looper (SBL),Pseudoplusia includens (Walker), and larvae parasitized byCopidosoma truncatellum (Dalman). SBL larvae, whether parasitized or not, consumed more foliage when fed “Ransom”. Unparasitized larvae reared on “81–327” had longer developmental times and suffered greater mortality than unparasitized larvae reared on “Ransom”. Parasitization of SBL larvae byC. truncatellum increased total foliage consumption of both soybean lines. Parasitized larvae reared on the resistant “81–327” weighed less and yielded fewer parasitoid adults.

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Résumé Des larves dePseudoplusia includens (Walker) parasitées ou non parCopidosoma truncatellum (Dalman) ont été nourries des feuilles de deux lignées de soja [Glycine max (L.) Merrill] l'une, GAT “81–327” résistante et l'autre, “Ransom” sensible. Les larves deP. includens qu'elles soient parasitées ou non consommaient plus de feuillage lorsqu'elles étaient nourries de la lignée “Ransom”. Les larves non parasitées élevées sur “81–327” avaient un cycle de développement beaucoup plus long et un taux de mortalité beaucoup plus élevé que les larves non parasitées élevées comparativement sur feuilles de “Ransom”. Par contre, les larves parasitées manifestaient une consommation accrue du feuillage des deux lignées de soja. Les larves parasitées élevées sur les feuilles de la variété résistante GAT “81–327” pesaient moins et produisaient moins également de parasites adultes.

     

A neural model for conditioned avoidance learning

A mathematical model for learning of a conditioned avoidance behavior is presented. An identification of the net excitation of a neural model (Rashevsky, N., 1960.Mathematical Biophysics. Vol. II. New York: Dover Publications, Inc.) with the instantaneous probability of response is introduced and its usefulness in discussing block-trial learning performances in the conditioned avoidance situation is outlined for normal and brain-operated animals, using experimental data collected by the author. Later, the model is applied to consecutive trial learning and connection is made with the approach of H. D. Landahl (1964. “An Avoidance Learning Situation. A Neural Net Model.”Bull. Math. Biophysics,26, 83–89; and 1965, “A Neural Net Model for Escape Learning.”Bull. Math. Biophysics,27, Special Edition, 317–328) wherein lie further data with which the model can be compared.     

On the utilization of food by planktophage fishes

Making some plausible assumptions about the over-all mechanism of food catching and consumption by fishes and evaluating in the light of those assumptions some available experimental data, it is possible to calculate from those data the variation of several important factors with the concentration of food. The factors considered are: total rate of metabolism, total diurnal energy expenditure in the process of feeding, average number of hours per day during which the fish feeds, average length of path traveled by a fish per day, and the so-called “energetic coefficient of growth.” A possible relation with the work of N. Rashevsky (Bull. Math. Biophysics,20, 299–308, 1959) is discussed.     

Psychoontogeny and psychophylogeny: Bernhard Rensch’s (1900–1990) selectionist turn through the prism of panpsychistic identism

Toward the end of the 1930s, Bernhard Rensch (1900–1990) turned from Lamarckism and orthogenesis to selectionism and became one of the key figures in the making of the Synthetic Theory of Evolution (STE). He contributed to the Darwinization of biological systematics, the criticism of various anti-Darwinian movements in the German lands, but more importantly founded a macroevolutionary theory based on Darwinian gradualism. In the course of time, Rensch’s version of the STE developed into an all-embracing metaphysical conception based on a kind of Spinozism. Here we approach Rensch’s “selectionist turn” by outlining its context, and by analyzing his theoretical transformation. We try to reconstruct the immanent logic of Rensch’s evolution from a “Lamarckian Synthesis” to a “Darwinian Synthesis”. We will pay close attention to his pre-Darwinian works, because this period has not been treated in detail in English before. We demonstrate an astonishing continuity in topics, methodology, and empirical generalizations despite the shift in Rensch’s views on evolutionary mechanisms. We argue that the continuity in Rensch’s theoretical system can be explained, at last in part, by the guiding role of general methodological principles which underlie the entire system, explicitly or implicitly. Specifically, we argue that Rensch’s philosophy became an asylum for the concept of orthogenesis which Rensch banned from evolutionary theory. Unable to explain the directionality of evolution in terms of empirically based science, he “pre-programmed” the occurrence of human-level intelligence by a sophisticated philosophy combined with a supposedly naturalistic evolutionary biology.

Explaining how and explaining why: developmental and evolutionary explanations of dominance

There have been two different schools of thought on the evolution of dominance. On the one hand, followers of Wright [Wright S. 1929. Am. Nat. 63: 274–279, Evolution: Selected Papers by Sewall Wright, University of Chicago Press, Chicago; 1934. Am. Nat. 68: 25–53, Evolution: Selected Papers by Sewall Wright, University of Chicago Press, Chicago; Haldane J.B.S. 1930. Am. Nat. 64: 87–90; 1939. J. Genet. 37: 365–374; Kacser H. and Burns J.A. 1981. Genetics 97: 639–666] have defended the view that dominance is a product of non-linearities in gene expression. On the other hand, followers of Fisher [Fisher R.A. 1928a. Am. Nat. 62: 15–126; 1928b. Am. Nat. 62: 571–574; Bürger R. 1983a. Math. Biosci. 67: 125–143; 1983b. J. Math. Biol. 16: 269–280; Wagner G. and Burger R. 1985. J. Theor. Biol. 113: 475–500; Mayo O. and Reinhard B. 1997. Biol. Rev. 72: 97–110] have argued that dominance evolved via selection on modifier genes. Some have called these “physiological” versus “selectionist,” or more recently [Falk R. 2001. Biol. Philos. 16: 285–323], “functional,” versus “structural” explanations of dominance. This paper argues, however, that one need not treat these explanations as exclusive. While one can disagree about the most likely evolutionary explanation of dominance, as Wright and Fisher did, offering a “physiological” or developmental explanation of dominance does not render dominance “epiphenomenal,” nor show that evolutionary considerations are irrelevant to the maintenance of dominance, as some [Kacser H. and Burns J.A. 1981. Genetics 97: 639–666] have argued. Recent work [Gilchrist M.A. and Nijhout H.F. 2001. Genetics 159: 423–432] illustrates how biological explanation is a multi-level task, requiring both a “top-down” approach to understanding how a pattern of inheritance or trait might be maintained in populations, as well as “bottom-up” modeling of the dynamics of gene expression.     

Method as a Function of “Disciplinary Landscape”: C.D. Darlington and Cytology, Genetics and Evolution, 1932–1950

This article considers the reception of British cytogeneticist C.D. Darlington’s controversial 1932 book, Recent Advances in Cytology. Darlington’s cytogenetic work, and the manner in which he made it relevant to evolutionary biology, marked an abrupt shift in the status and role of cytology in the life sciences. By focusing on Darlington’s scientific method – a stark departure from anti-theoretical, empirical reasoning to a theoretical and speculative approach based on deduction from genetic first principles – the article characterises the relationships defining the “disciplinary landscape” of the life sciences of the time, namely those between cytology, genetics, and evolutionary theory.     

A contribution to the mathematical biology of automobile driving: II. Passing as a case of psychophysical discrimination

The decision to pass or not to pass in view of an oncoming car is considered as a case of comparative judgment in which it is to be decided whether the time it will take to pass safely is greater or less than the time it will take to collide with the oncoming car. H. D. Landahl's well-known theory of psychophysical discrimination is used, and it is assumed that the “distracting stimuli” considered previously (Rashevsky, 1959,Bull. Math. Biophysics,21, 375–85) tend to increase the standard deviation of Landahl's fluctuation function. Effects of the “distracting stimuli” on the threshold of the neuroelements in Landahl's circuit are also considered. On this basis an expression is derived which gives the probability of a collision accident in passing as a function of the “distracting stimuli.”     

The “Cycle of Life” in Ecology: Sergei Vinogradskii’s Soil Microbiology, 1885–1940

Historians of science have attributed the emergence of ecology as a discipline in the late nineteenth century to the synthesis of Humboldtian botanical geography and Darwinian evolution. In this essay, I begin to explore another, largely neglected but very important dimension of this history. Using Sergei Vinogradskii’s career and scientific research trajectory as a point of entry, I illustrate the manner in which microbiologists, chemists, botanists, and plant physiologists inscribed the concept of a “cycle of life” into their investigations. Their research transformed a longstanding notion into the fundamental approaches and concepts that underlay the new ecological disciplines that emerged in the 1920s. Pasteur thus joins Humboldt as a foundational figure in ecological thinking, and the broader picture that emerges of the history of ecology explains some otherwise puzzling features of that discipline – such as its fusion of experimental and natural historical methodologies. Vinogradskii’s personal “cycle of life” is also interesting as an example of the interplay between Russian and Western European scientific networks and intellectual traditions. Trained in Russia to investigate nature as a super-organism comprised of circulating energy, matter, and life; over the course of five decades – in contact with scientists and scientific discourses in France, Germany, and Switzerland – he developed a series of research methods that translated the concept of a “cycle of life” into an ecologically conceived soil science and microbiology in the 1920s and 1930s. These methods, bolstered by his authority as a founding father of microbiology, captured the attention of an international network of scientists. Vinogradskii’s conceptualization of the “cycle of life” as chemosynthesis, autotrophy, and global nutrient cycles attracted the attention of ecosystem ecologists; and his methods appealed to practitioners at agricultural experiment stations and microbiological institutes in the United States, Western Europe, and the Soviet Union.     

Laminin provides a better substrate than fibronectin for attachment,growth, and differentiation of 1003 embryonal carcinoma cells

Summary Culture of cells in hormonally defined media has allowed (a) the demonstration of physiological responses from cells usually unable to express them in vitro and (b) the study of the effects on growth and differentiation of diffusible factors and attachment factors. The embryonal carcinoma line 1003 forms multidifferentiated tumors in vivo but is unable to differentiate in vitro when grown in serum-containing medium. In a defined medium containing insulin, transferrin, selenium, and fibronectin as attachment factors, 1003 cells grow for several generations and differentiate into neurons and embryonic mesenchyme (Darmon et al., 1981, Dev. Biol. 85: 463–473). In the present work the effects of fibronectin and laminin were compared. In the presence of laminin the cells attached and spread better, grew faster, and could be plated at lower densities. Neurite extension was also better under these conditions and most importantly, it was found that laminin induced an important formation of muscular tissue when the cells had been seeded at low densities. Multinucleated myotubes could be stained with antibodies directed against embryonic muscular myosin. Coating the dishes with polylysine or adding FGF or serum-spreading factor to the medium allowed growth of low-density cultures with fibronectin instead of laminin but muscular differentiation was not detected under these conditions. Addition of fibronectin to laminin-containing medium did not inhibit muscular differentiation. Presented in the symposium on Plant and Animal Physiology in Vitro at the 33rd Annual Meeting of the Tissue Culture Association, San Diego, California, June 6–10, 1982. This research was supported in part by grants from the “Centre National de la Recherche Scientifique” (LA 269), the “Délégation Générale à la Recherche Scientifique et Technique,” the Fondation pour la Recherche Médicale Fran?aise,” the “Institut National de la Santé et de la Recherche Medicale,” the “Ligue Nationale Fran?aise centre le Cancer,” and the “Fondation André Meyer.” This symposium was supported in part by the following organizations: Bellco Glass, Inc., California Branch of the Tissue Culture Association, Collaborative Research, Hana Media, Hybridtech, K C Biological, Inc., and Millipore Corporation.     

Wallace’s Other Line: Human Biogeography and Field Practice in the Eastern Colonial Tropics

This paper examines how the 19th-century British naturalist Alfred Russel Wallace used biogeographical mapping practices to draw a boundary line between Malay and Papuan groups in the colonial East Indies in the 1850s. Instead of looking for a continuous gradient of variation between Malays and Papuans, Wallace chose to look for a sharp discontinuity between them. While Wallace’s “human biogeography” paralleled his similar project to map plant and animal distributions in the same region, he invoked distinctive “mental and moral” features as more decisive than physical ones. By following Wallace in the field, we can see his field mapping practices in action – how he conquered the problem of local particularity in the case of human variation. His experiences on the periphery of expanding European empires, far from metropolitan centers, shaped Wallace’s observations in the field. Taking his cues from colonial racial categories and his experiences collaborating with local people in the field, Wallace constructed the boundary line between the Malay and Papuan races during several years of work in the field criss-crossing the archipelago as a scientific collector. This effort to map a boundary line in the field was a bold example of using the practices of survey science to raise the status of field work by combining fact gathering with higher-level generalizing, although the response back in the metropole was less than enthusiastic. Upon his return to Britain in the 1860s, Wallace found that appreciation for observational facts he had gathered in the field was not accompanied by agreement with his theoretical interpretations and methods for doing human biogeography.     

Terahertz vibrational properties of water nanoclusters relevant to biology

Water nanoclusters are shown from first-principles calculations to possess unique terahertz-frequency vibrational modes in the 1–6 THz range, corresponding to O–O–O “bending,” “squashing,” and “twisting” “surface” distortions of the clusters. The cluster molecular-orbital LUMOs are huge Rydberg-like “S,” “P,” “D,” and “F” orbitals that accept an extra electron via optical excitation, ionization, or electron donation from interacting biomolecules. Dynamic Jahn–Teller coupling of these “hydrated-electron” orbitals to the THz vibrations promotes such water clusters as vibronically active “structured water” essential to biomolecular function such as protein folding. In biological microtubules, confined water-cluster THz vibrations may induce their “quantum coherence” communicated by Jahn–Teller phonons via coupling of the THz electromagnetic field to the water clusters’ large electric dipole moments.     

Evolutionary Morphology in Belgium

In historical literature, Edouard van Beneden (1846–1910) is mostly remembered for his cytological discoveries. Less well known, however, is that he also introduced evolutionary morphology – and indeed evolutionary theory as such – in the Belgian academic world. The introduction of this research programme cannot be understood without taking both the international and the national context into account. It was clearly the German example of the Jena University that inspired van Beneden in his research interests. The actual launch of evolutionary morphology at his University of Liège was, however, also connected with the dynamic of Belgian university reforms and the local rationale of creating a research “school.” Thanks to his networks, his mastering of the rhetoric of the “new” biology, his low ideological profile and his capitalising on the new academic élan in late-19th century Belgium, van Beneden managed to turn his programme into a local success from the 1870s onwards. Two decades later, however, the conceptual underpinnings of evolutionary morphology came under attack and the “Van Beneden School” lost much of its vitality. Despite this, van Beneden’s evolutionary morphology was prototypical for the research that was to come. He was one of the first scientific heavyweights in Belgium to turn the university laboratory into a centre of scientific practice and the hub of a research school.     

The Preface to Darwin’s <Emphasis Type="Italic">Origin of Species</Emphasis>: The Curious History of the “Historical Sketch” *

Almost any modern reader’s first encounter with Darwin’s writing is likely to be the “Historical Sketch,” inserted by Darwin as a preface to an early edition of the Origin of Species, and having since then appeared as the preface to every edition after the second English edition. The Sketch was intended by him to serve as a short “history of opinion” on the species question before he presented his own theory in the Origin proper. But the provenance of the “Historical Sketch” is somewhat obscure. Some things are known about its production, such as when it first appeared and what changes were made to it between its first appearance in 1860 and its final form, for the fourth English edition, in 1866. But how it evolved in Darwin’s mind, why he wrote it at all, and what he thought he was accomplishing by prefacing it to the Origin remain questions that have not been carefully addressed in the scholarly literature on Darwin. I attempt to show that Darwin’s various statements about the “Historical Sketch,” made primarily to several of his correspondents between 1856 and 1860, are somewhat in conflict with one another, thus making problematic a satisfactory interpretation of how, when, and why the Sketch came to be. I also suggest some probable resolutions to the several difficulties. How Darwin came to settle on the title “Historical Sketch” for the Preface to the Origin is not certain, but a guess may be ventured. When he first submitted the text to Asa Gray in February 1860 he called it simply “Preface Contributed by the Author to this American Edition” (Burkhardt et al., eds., vol. 8, 1993, p. 572; the collected correspondence is hereafter cited as CCD). In fact he had thought of it as being properly called a Preface much earlier, perhaps as early as 1856, as will be seen in what follows. It came to be called “An Historical Sketch of the Recent Progress of Opinion on the Origin of Species” only in the third English edition, April 1861. This is the title it retained thereafter, with the exception of an addition to the title in the sixth English edition, “Previously to the Publication of the First Edition of this Work” (Peckham, 1959, pp. 20, 59). The word “sketch,” on the other hand was one of two words Darwin commonly used in private correspondence to refer to the book that would later become the Origin, the other word being “Abstract,” and both signifying that Darwin thought of the work as being a resume rather than a full-fledged study (e.g., letter to J.D. Hooker, May 9 1856, CCD vol. 6 p. 106; letter to Baden Powell January 18 1860, CCD vol. 8 p. 41; letter to Lyell 25 June 1858, CCD v. 7, 1991, pp. 117–8; letter to Lyell May 1856, CCD, v. 6 p. 100). The most likely source of the title “Historical Sketch” for Darwin’s Preface is Charles Lyell’s Principles of Geology in which, beginning with the third edition (1834), Lyell added titles to his chapters, calling chapters 2–4 “Historical Sketch of the Progress of Geology” (Secord, in Lyell [1997], p. xlvii; for other uses by Lyell of this expression, cf. Porter, 1976, p. 95; idem 1982, p. 38; and Lyell, 1830 [1990], p. 30). Further parallels between Lyell’s Introduction and Darwin’s “Historical Sketch” in terms of content and strategy are suggested below.     

Some considerations on relational equilibria

A previous study (Bull. Math. Biophysics,31, 417–427, 1969) on the definitions of stability of equilibria in organismic sets determined byQ relations is continued. An attempt is made to bring this definition into a form as similar as possible to that used in physical systems determined byF-relations. With examples taken from physics, biology and sociology, it is shown that a definition of equilibria forQ-relational systems similar to the definitions used in physics can be obtained, provided the concept of stable or unstable structures of a system determined byQ-relations is considered in a probabilistic manner. This offers an illustration of “fuzzy categories,” a notion introduced by I. Bąianu and M. Marinescu (Bull. Math. Biophysics,30, 625–635, 1968), in their paper on organismic supercategories, which is designed to provide a mathematical formalism for Rashevsky's theory of Organismic Sets (Bull. Math. Biophysics,29, 389–393, 1967;30, 163–174, 1968;31, 159–198, 1969). A suggestion is made for a method of mapping the abstract discrete space ofQ-relations on a continuum of variables ofF-relations. Problems of polymorphism and metamorphosis, both in biological and social organisms, are discussed in the light of the theory.     

Multiple relational equilibria: Polymorphism,metamorphosis and other possibly similar phenomena

In a preceding paper (Rashevsky, 1969. “Outline of a Unified Approach to Physics, Biology and Sociology.”Bulletin of Mathematical Biophysics,31, 159–198) certain isomorphisms between biological and social systems on the one hand and physical systems on the other were studied. The notion or relational forces, of which ordinary physical forces are a particular case, was introduced. In the present paper an attempt is made to establish analogies between stable equilibria in physical systems, equilibria due to physical forces, and stable equilibria in biological and social systems which are due to purely relational forces. The notion of relational forces causing multiple equilibria similar to multiple equilibria in some physical systems is studied, and it is outlined how this notion may possibly help the understanding of such phenomena as polymorphism, metamorphosis and the existence of rudimentary organs or rudimentary functions.     

Agonistic and courtship behaviour patterns in the skink Chalcides viridanus (Fam. Scincidae) from Tenerife

There have been relatively few attempts to quantitatively describe behaviours in scincid lizards. Chalcides viridanus is a small body-sized skink endemic of Tenerife (Canary Islands). We describe and quantify 18 behaviour patterns (both social and agonistic) of this species, some of which have not been described before for other scincids. Video recordings of male–male, female–female, and male–female interactions were made under laboratory conditions, with controlled light–dark cycle and temperature. We describe several agonistic and courtship behaviour patterns. Within the first context, we detected a new agonistic behaviour for a scincid, “Snout to body”, that appeared at the beginning of agonistic sequences; it consisted of each animal placing its snout in contact with the other individual’s lateral side of the body. The amplitude of head movement during “Head bobbing” was lower than that described for many other lizard species. Agonistic behaviours were shown in intrasexual staged encounters both within males and females. The comparison of behaviour patterns of both types of intrasexual encounters showed that females were more active, exhibiting significantly higher frequencies of behaviour than males. Specifically, females showed the “Snout to body” pattern more frequently than males. In male–female encounters we detected courtship and copulation patterns only in April, when males performed “Bites” and “Snout to body” directed at females.