Topology and life: In search of general mathematical principles in biology and sociology

Mathematical biology has hitherto emphasized the quantitative, metric aspects of the physical manifestations of life, but has neglected the relational or positional aspects, which are of paramount importance in biology. Although, for example, the processes of locomotion, ingestion, and digestion in a human are much more complex than in a protozoan, the general relations between these processes are the same in all organisms. To a set of very complicated digestive functions of a higher animal there correspond a few simple functions in a protozoan. In other words, the more complicated processes in higher organisms can be mapped on the simpler corresponding processes in the lower ones. If any scientific study of this aspect of biology is to be possible at all, there must exist some regularity in such mappings. We are, therefore, led to the following principle: If the relations between various biological functions of an organism are represented geometrically in an appropriate topological space or by an appropriate topological complex, then the spaces or complexes representing different organisms must be obtainable by a proper transformation from one or very fewprimordial spaces or complexes. The appropriate representation of the relations between the different biological functions of an organism appears to be a one-dimensional complex, or graph, which represents the “organization chart” of the organism. The problem then is to find a proper transformation which derives from this graph the graphs of all possible higher organisms. Both a primordial graph and a transformation are suggested and discussed. Theorems are derived which show that the basic principle of mapping and the transformation have a predictive value and are verifiable experimentally. These considerations are extended to relations within animal and human societies and thus indicate the reason for the similarities between some aspects of societies and organisms. It is finally suggested that the relation between physics and biology may lie on a different plane from the one hitherto considered. While physical phenomena are the manifestations of the metric properties of the four-dimensional universe, biological phenomena may perhaps reflect some local topological properties of that universe.     

Physics,biology, and sociology: A reappraisal

To the extent that all biological phenomena are perceivable only through their physical manifestations, it may be justified to assume that all biological phenomena will be eventually represented in terms of physics; perhaps not of present day physics, but of some “extended” form of it. However, even if this should be correct, it must be kept in mind that representing individual biological phenomena in terms of physics is not the same as deducing from known physical laws the necessity of biological phenomena. Drawing an analogy from pure mathematics, it is possible that while every biological phenomenon may be represented in terms of physics, yet biological statements represent a class of “undecidable” statements within the framework of physics. Such a conjecture is reinforced by the history of physics itself and illustrated on several examples. The 19th century physicists tried in vain todeduce electromagnetic phenomena from mechanical ones. A similar situation may exist in regard to biological and social sciences. Quite generally, the possibility of representing a class B phenomena in terms of class A phenomena does not imply that the phenomena of class B can be deduced from those of class A. The consequences of the above on the relation between physics, biology, and sociology are studied. A tentative postulational formulation of basic biological principles are given and some consequences are discussed. It is pointed out that not only can the study of biological phenomena throw light on some physical phenomena, but that the study of social phenomena may be of value for the understanding of the structures and functions of living organisms. The possibility of a sort of “socionics” is indicated.     

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.     

What is “Biological Physics”? A resource letter

The purpose of this resource letter is threefold: To attempt a refinement of the tenuous definition of the term “Biological Physics”. To do this via a compendium, albeit inexhaustive and incomplete, of materials which might appropriately be labelled biological physics. To provide a useful introduction to the learning resources in biological physics for college students and their professors.     

“Bottom-up” approach for implementing nano/microstructure using biological and chemical interactions

The “Bottom-up” approach for implementing nano/microstructure using biological self-assembled systems has been investigated with tremendous interest by many researchers in the field of medical diagnostics, material synthesis, and nano/microelectronics. As a result, the techniques for achieving these systems have been extensively explored in recent years. The developed or developing techniques are based on many interdisciplinary areas such as biology, chemistry, physics, electrical engineering, mechanical engineering, and so on. In this paper, we review the fundamentals behind the self-assembly concepts and describe the state of art in the biological and chemical self-assembled systems for the implementation of nano/microstructures. These structures described in the paper can be applied to the implementation of hybrid biosensors, biochip, novel bio-mimetic materials, and nano/microelectronic devices.     

Nicolas Rashevsky's Mathematical Biophysics

This paper explores the work of Nicolas Rashevsky, a Russian émigré theoretical physicist who developed a program in “mathematical biophysics” at the University of Chicago during the 1930s. Stressing the complexity of many biological phenomena, Rashevsky argued that the methods of theoretical physics – namely mathematics – were needed to “simplify” complex biological processes such as cell division and nerve conduction. A maverick of sorts, Rashevsky was a conspicuous figure in the biological community during the 1930s and early 1940s: he participated in several Cold Spring Harbor symposia and received several years of funding from the Rockefeller Foundation. However, in contrast to many other physicists who moved into biology, Rashevsky's work was almost entirely theoretical, and he eventually faced resistance to his mathematical methods. Through an examination of the conceptual, institutional, and scientific context of Rashevsky's work, this paper seeks to understand some of the reasons behind this resistance. This revised version was published online in July 2006 with corrections to the Cover Date.     

From “Us vs. Them” to “Shared Risk”: Can Animals Help Link Environmental Factors to Human Health?

Linking human health risk to environmental factors can be a challenge for clinicians, public health departments, and environmental health researchers. While it is possible that nonhuman animal species could help identify and mitigate such linkages, the fields of animal and human health remain far apart, and the prevailing human health attitude toward disease events in animals is an “us vs. them” paradigm that considers the degree of threat that animals themselves pose to humans. An alternative would be the development of the concepts of animals as models for environmentally induced disease, as well as potential “sentinels” providing early warning of both noninfectious and infectious hazards in the environment. For such concepts to truly develop, critical knowledge gaps need to be addressed using a “shared risk” paradigm based on the comparative biology of environment–host interactions in different species.     

Against reduction

In Molecular Models: Philosophical Papers on Molecular Biology, Sahotra Sarkar presents a historical and philosophical analysis of four important themes in philosophy of science that have been influenced by discoveries in molecular biology. These are: reduction, function, information and directed mutation. I argue that there is an important difference between the cases of function and information and the more complex case of scientific reduction. In the former cases it makes sense to taxonomise important variations in scientific and philosophical usage of the terms “function” and “information”. However, the variety of usage of “reduction” across scientific disciplines (and across philosophy of science) makes such taxonomy inappropriate. Sarkar presents reduction as a set of facts about the world that science has discovered, but the facts in question are remarkably disparate; variously semantic, epistemic and ontological. I argue that the more natural conclusion of Sarkar’s analysis is eliminativism about reduction as a scientific concept.     

Organismic sets: Outline of a general theory of biological and social organisms

The discussion as to whether societies are organisms andvice versa has been going on for a long time. The question is meaningless unless a clear definition of the term “organism” is made. Once such a definition is made, the question may be answered by studying whether there exists any relational isomorphism between what the biologist calls an organism and what the sociologist calls society. Such a study should also include animal societies studied by ecologists. Both human and animal societies are sets of individuals together with certain other objects which are the products of their activities. A multicellular organism is a set of cells together with some products of their activities. A cell itself may be regarded as a set of genes together with the products of their activities because every component of the cell is either directly or indirectly the result of the activities of the genes. Thus it is natural to define both biological and social organisms as special kinds of sets. A number of definitions are given in this paper which define what we call here organismic sets. Postulates are introduced which characterize such sets, and a number of conclusions are drawn. It is shown that an organismic set, as defined here, does represent some basic relational aspects of both biological organisms and societies. In particular a clarification and a sharpening of the Postulate of Relational Forces given previously (Bull. Math. Biophysics,28, 283–308, 1966) is presented. It is shown that from the basic definitions and postulates of the theory of organismic sets, it folows that only such elements of those sets will aggregate spontaneously, which are not completely “specialized” in the performance of only one activity. It is further shown that such “non-specialized” elements undergo a process of specialization, and as a result of it their spontaneous aggregation into organismic sets becomes impossible. This throws light on the problem of the origin of life on Earth and the present absence of the appearance of life by spontaneous generation. Some applications to problems of ontogenesis and philogenesis are made. Finally the relation between physics, biology, and sociology is discussed in the light of the theory of organismic sets.     

Living Precisely in Fin-de-Siècle Vienna

Vienna’s Institute of Experimental Biology, better known as the Vivarium, helped pioneer the quantification of experimental biology from 1903 to 1938. Among its noteable scientists were the director Hans Przibram and his brother Karl (a physicist), Paul Kammerer, Eugen Steinach, Paul Weiss, and Karl Frisch. The Vivarium’s scientists sought to derive laws describing the development of the individual organism and its relationship to the environment. Unlike other contemporary proponents of biological laws, however, these researchers created an explicitly anti-deterministic science. By “laws” they meant statistical regularities or “patterns.” They interpreted their experimental results in ways that forged a “third way” between determinism and pure spontaneity, aiming to capture the complexity of the interaction between the organism and its environment. This common feature of their research was made possible by the availability at the Vivarium of the latest in climate-control technology and of methods borrowed from statistical physics. The deeper roots of this search for a “third way” lay, I suggest, in the shared educational, social, and aesthetic experiences of the laboratory’s workers.     

Protention and retention in biological systems

This article proposes an abstract mathematical frame for describing some features of cognitive and biological time. We focus here on the so called “extended present” as a result of protentional and retentional activities (memory and anticipation). Memory, as retention, is treated in some physical theories (relaxation phenomena, which will inspire our approach), while protention (or anticipation) seems outside the scope of physics. We then suggest a simple functional representation of biological protention. This allows us to introduce the abstract notion of “biological inertia”.     

Physics,biology and sociology: II. Suggestion for a synthesis

In continuation of previous studies (Bull. Math. Biophysics,28, 283–308; 655–661, 1966;29, 139–152, 1967) it is shown that the difference between the “metric” aspects of physics and the “relational” aspects of biological and social sciences disappear by accepting the broader definition of “relation”, such as that given in mathematics and logic. A conceptual superstructure then becomes possible from which all three branches of knowledge may be derived, though none of them can be derived from the others.     

Topological Patterns in Metazoan Evolution and Development

Topological patterns in the development and evolution of metazoa, from sponges to chordates, are considered by means of previously elaborated methodology, with the genus of the surface used as a topological invariant. By this means metazoan morphogenesis may be represented as topological modification(s) of the epithelial surfaces of an animal body. The animal body surface is an interface between an organism and its environment, and topological transformations of the body surface during metazoan development and evolution results in better distribution of flows to and from the external medium, regarded as the source of nutrients and oxygen and the sink of excreta, so ensuring greater metabolic intensity. In sponges and some Cnidaria, the increase of this genus up to high values and the shaping of topologically complicated fractal-like systems are evident. In most Bilateria, a stable topological pattern with a through digestive tube is formed, and the subsequent topological complications of other systems can also appear. The present paper provides a topological interpretation of some developmental events through the use of well-known mathematical concepts and theorems; the relationship between local and global orders in metazoan development, i.e., between local morphogenetic processes and integral developmental patterns, is established. Thus, this methodology reveals a “topological imperative”: A certain set of topological rules that constrains and directs biological morphogenesis.     

Does the Segregation of Evolution in Biology Textbooks and Introductory Courses Reinforce Students’ Faulty Mental Models of Biology and Evolution?

The well-established finding that substantial confusion and misconceptions about evolution and natural selection persist after college instruction suggests that these courses neither foster accurate mental models of evolution’s mechanisms nor instill an appreciation of evolution’s centrality to an understanding of the living world. Our essay explores the roles that introductory biology courses and textbooks may play in reinforcing undergraduates’ pre-existing, faulty mental models of the place of evolution in the biological sciences. Our content analyses of the three best-selling introductory biology textbooks for majors revealed the conceptual segregation of evolutionary information. The vast majority of the evolutionary terms and concepts in each book were isolated in sections about evolution and diversity, while remarkably few were employed in other sections of the books. Standardizing the data by number of pages per unit did not alter this pattern. Students may fail to grasp that evolution is the unifying theme of biology because introductory courses and textbooks reinforce such isolation. Two goals are central to resolving this problem: the desegregation of evolution as separate “units” or chapters and the active integration of evolutionary concepts at all levels and across all domains of introductory biology.     

Dimensional analysis in mathematical biology. II

The application of dimensional analysis in biology is further illustrated by functional equations composed of dimensionless numbers and dealing with renal physiology, lung physiology and plant leaf shape. Dimensional variables and dimensionless numbers are examined from the viewpoint of numerical invariant properties of a certain physical system. Utilization of the method for problems such as design of an artificial kidney is considered briefly. A tabulation of variables useful in biology is given, with suggestions for a number of new dimensional entities. A continuation of the list of dimensionless invariants from Part I (Bull. Math. Biophysics,23, 355–376, 1961) is provided and includes terms pertaining to general physiology, geometric growth, metabolism, ecological interactions, muscle kinetics and other areas. It is pointed out that use of dimensionless ratios (similarity criteria) makes possible a direct comparison of form or shape factors and relative growth ratios with a variety of physical ratios, through the use of functional equations containing only dimensionless entities. Organismal similarity during growth and development, and between genetically related species, may be analyzed in terms of “automodel” or “self-similar” systems governed by certain dimensionless invariants. Tables of biological variables and dimensionless groupings are included.     

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.     

Theoretical approaches to holistic biological features: Pattern formation,neural networks and the brain-mind relation

The topic of this article is the relation between bottom-up and top-down, reductionist and “holistic” approaches to the solution of basic biological problems. While there is no doubt that the laws of physics apply to all events in space and time, including the domains of life, understanding biology depends not only on elucidating the role of the molecules involved, but, to an increasing extent, on systems theoretical approaches in diverse fields of the life sciences. Examples discussed in this article are the generation of spatial patterns in development by the interplay of autocatalysis and lateral inhibition; the evolution of integrating capabilities of the human brain, such as cognition-based empathy; and both neurobiological and epistemological aspects of scientific theories of consciousness and the mind.     

Chain processes and their biophysical applications: Part I. General theory

A mathematical theory applicable to the biological effects of radiations as chain processes is developed. The theory may be interpreted substantially as a “hit theory” involving the concepts of “sensitive volume” or “target area”. The variability of the sensitivity of the organism to the radiation and its capacity of recovery between single hits is taken into account. It is shown that in a continuous irradiation of a biological aggregate in which the effect of each single hit cannot be observed, recovery and variation of sensitivity are formally equivalent to each other so that a discrimination between these two phenomena is possible only by discontinuous irradiation or by using different radiation intensities. Methods for the calculation of the “number of hits” and for the determination of the kinetics of the processes from “survival curves” or similar experimental data are given. The relation between the recovery and the Bunsen-Roscoe law is discussed. The case in which the injury of the organism is dependent on the destruction of more than one “sensitive volume” is also considered.     

The Extended Synthesis: Something Old, Something New

The eclipse of Darwinism began to end in the 1980s and hangs in the balance today. We need an Extended Synthesis, using “extension” metaphorically. We must extend back in time to recover important aspects of Darwinism that were set aside, and then lost during neo-Darwinism, then move forward beyond neo-Darwinism to encompass new data and concepts. The most comprehensive framework for the Extended Synthesis is the Major Transitions in Evolution. The Extended Synthesis rests comfortably within a philosophical perspective in which biology does not need to be connected with other areas of science in order to justify itself. I am attracted to an older concept in which biology needs a covering law to connect it with the rest of the natural sciences. Darwin implicated a “higher law,” but did not specify it. If we can elucidate that law, the Extended Synthesis will become the Unified Theory of Biology called for by Brooks and Wiley 25 years ago.     

Helix-mediated protein–protein interactions as targets for intervention using foldamers

Protein–protein interactions (PPIs) play a central role in virtually all biological processes and have been the focus of intense investigation from structural molecular biology to cell biology for the majority of the last two decades and, more recently, are emerging as important targets for pharmaceutical intervention. A common motif found at the interface of PPIs is the α-helix, suggesting that, in the same way as the “lock and key” model has evolved for competitive inhibition of enzymes, it should be possible to elaborate “rule-based” approaches for inhibition of helix-mediated PPIs. This review will describe the biological function and structural features of a series of representative helix-mediated PPIs and discuss approaches that are being developed to target these interactions with small molecules that employ non-natural amino acids.