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.     

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.     

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.     

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.     

Stieltjes integration and differential geometry: A model for enzyme recognition,discrimination, and catalysis

A model for enzymic catalysis is presented using the mathematical theories of differential geometry and Stieltjes integration. The Stieltjesintegrator is a complex-valued function of bounded variation which represents the curvature and torsion, hence the conformation, of the backbone of an enzyme molecule. Theintegrand is a complex-valued continuous function which describes the shape of the surface of a substrate molecule. We postulate that enzyme-substrate interactions correspond to evaluations of Stieltjes integrals, and that observables of enzymic catalysis correspond to projections. Results from the mathematical theory of the Stieltjes integral are discussed together with their biological interpretations. We contrast the difference between structural and functional proteins, and construct analogues of enzyme cofactors, modifications, and regulation. Various techniques of locating the active site on enzymes are also given. We construct a total variation metric, which is particularly useful for detecting similarities among proteins. An examination on the many different modes of convergence of mathematical functions representing biological molecules leads to a mathematical statement of the fundamental dogma of molecular biology, that ‘structure implies function’. Similar arguments also result in the converse statement ‘function dictates structure’, which is a basic premise of relational biology. Stepped-helical approximations of the backbone space curves of enzymes provide a concrete computational tool with which to calculate the Stieltjes integrals that model enzymic catalysis, by replacing the integral with a finite series. The duality between enzymes and substrates (that they aremeters ‘observing’ one another) is shown to be a consequence of the mathematical duality of Banach spaces. The Stieltjes integrals of enzyme-substrate interactions are hence shown to be bounded bilinear functionals. The mechanism of enzymic catalysis, the transformation from substrate to product, is also formulated in the Stieltjes integration context via the mathematical theory of adjoints. The paper closes with suggestions for generalizations, prospects for future studies, and a review of the correspondence between mathematical and biological concepts.     

OpenFLUX: efficient modelling software for 13C-based metabolic flux analysis

Background  

The quantitative analysis of metabolic fluxes, i.e., in vivo activities of intracellular enzymes and pathways, provides key information on biological systems in systems biology and metabolic engineering. It is based on a comprehensive approach combining (i) tracer cultivation on ¹³C substrates, (ii) ¹³C labelling analysis by mass spectrometry and (iii) mathematical modelling for experimental design, data processing, flux calculation and statistics. Whereas the cultivation and the analytical part is fairly advanced, a lack of appropriate modelling software solutions for all modelling aspects in flux studies is limiting the application of metabolic flux analysis.     

Equivalence of the theories of nervous excitation of Hill,Monnier, and Rashevsky

New and direct proofs of the equivalence of the three theories of nervous excitation are developed, and by the introduction of theHauptnutzzeit as the unit of time and a special unit for the measurement of the excitatory state, a normal form for Rashevsky's equations is obtained which exhibits clearly the relations among the various independent parameters.     

A note on Rashevsky's theorem about point-bases in topological biology

A graphG may have more than one point-baseB _G. In a primordial graphP of Rashevsky's (1954) TransformationT, some of the pointbases may consist of nonspecialized points only, and some other pointbases may contains specialized points. In this case, Rashevsky's Theorem (1955a) on point-bases may not hold. The Theorem is certainly true ifall point-bases ofP consists of nonspecialized points. A rigorous proof is given. Some results are derived for the more general case, when point-bases include both kinds of points. A general expression for the point-base ratio of the transformed graphP(T) is obtained. It is shown that with some biologically plausible assumptions Rashevsky's interpretation of the point-base ratio and his conclusions are still true. A few simple Theorems on point-bases of graphs are included in this work.     

Constraints-based genome-scale metabolic simulation for systems metabolic engineering

Random mutagenesis and selection approaches used traditionally for the development of industrial strains have largely been complemented by metabolic engineering, which allows purposeful modification of metabolic and cellular characteristics by using recombinant DNA and other molecular biological techniques. As systems biology advances as a new paradigm of research thanks to the development of genome-scale computational tools and high-throughput experimental technologies including omics, systems metabolic engineering allowing modification of metabolic, regulatory and signaling networks of the cell at the systems-level is becoming possible. In silico genome-scale metabolic model and its simulation play increasingly important role in providing systematic strategies for metabolic engineering. The in silico genome-scale metabolic model is developed using genomic annotation, metabolic reactions, literature information, and experimental data. The advent of in silico genome-scale metabolic model brought about the development of various algorithms to simulate the metabolic status of the cell as a whole. In this paper, we review the algorithms developed for the system-wide simulation and perturbation of cellular metabolism, discuss the characteristics of these algorithms, and suggest future research direction.     

Molecular analysis of recombination sites within the immunoglobulin heavy chain locus of the rabbit

Previously, recombinations involving genes of the rabbit immunoglobulin heavy chain locus have been documented serologically. These data indicated that the sites at which the causative recombination events occurred could have been anywhere from within the V _H gene cluster up to, or 3 of, C. Since these sites could not be localized further by serological methods, we attempted to do this using techniques of molecular biology. DNAs from homozygous recombinant rabbits and from the appropriate non-recombinant parental haplotypes were characterized using Southern blots hybridized with a panel of probes derived from cloned regions of the rabbit immunoglobulin heavy chain gene complex. In all three recombinants, the site was downstream of the entireV _H cluster and upstream of the J _Hcluster within an 50 kilobase (kb) egion containing expanses of repetitive-sequence DNA as well as D _H genes. D _H-specific probes further showed that in two of the recombinants, the recombination appears to have occurred within or 5 of D _H1 and 5 of D _H2 genes; in the third it occurred 3 of the D _H2 genes but at least 5 kb 5 of the J _H region. Address for correspondence and offprint request to: R. G. Mage.     

Discussion: Turing's theory of morphogenesis—Its influence on modelling biological pattern and form

The evolution of spatial pattern is a central issue in developmental biology. Turing's (Phil. Trans. R. Soc. Lond. B237, 37–72, 1952) chemical theory of morphogenesis is a seminal contribution. In this talk I give a personal and necessarily limited view of its impact on mathematical and developmental biology. I briefly describe some of the interesting mathematical aspects of Turing's reaction-diffusion mechanism and discuss some of the different models which Turing's vision inspired. The emphasis throughout is on the practical biological applications of the various theories.     

Abstract representations of biological systems in supercategories

An abstract representation of biological systems from the standpoint of the theory of supercategories is presented. The relevance of such representations forG-relational biologies is suggested. In section A the basic concepts of our representation, that is class, system, supercategory and measure are introduced. Section B is concerned with the mathematical representation starting with some axioms and principles which are natural extensions of the current abstract representations in biology. Likewise, some extensions of the principle of adequate design are introduced in section C. Two theorems which present the connection between categories and supercategories are proved. Two other theorems concerning the dynamical behavior of biological and biophysical systems are derived on the basis of the previous considerations. Section D is devoted to a general study of oscillatory behavior in enzymic systems, some general quantitative relations being derived from our representation. Finally, the relevance of these results for a quantum theoretic approach to biology is discussed.     

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.     

1/f noise in membranes

The present situation of 1/f noise in the passage of ions across membranes is examined. A survey of biological and synthetic membranes is given at which a l/f frequency dependence has been observed in the spectrum of voltage or current fluctuations. Empirical relations and theories of 1/f noise in membranes are critically discussed. Supported by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 38 “Membranforschung”     

Model building and model checking for biochemical processes

A central claim of computational systems biology is that, by drawing on mathematical approaches developed in the context of dynamic systems, kinetic analysis, computational theory and logic, it is possible to create powerful simulation, analysis, and reasoning tools for working biologists to decipher existing data, devise new experiments, and ultimately to understand functional properties of genomes, proteomes, cells, organs, and organisms. In this article, a novel computational tool is described that achieves many of the goals of this new discipline. The novelty of this system involves an automaton-based semantics of the temporal evolution of complex biochemical reactions starting from the representation given as a set of differential equations. The related tools also provide ability to qualitatively reason about the systems using a propositional temporal logic that can express an ordered sequence of events succinctly and unambiguously. The implementation of mathematical and computational models in the Simpathica and XSSYS systems is described briefly. Several example applications of these systems to cellular and biochemical processes are presented: the two most prominent are Leibler et al.'s repressilator (an artificial synthesized oscillatory network), and Curto-Voit-Sorribas-Cascante's purine metabolism reaction model.     

An integrated cell‐free metabolic platform for protein production and synthetic biology

Cell‐free systems offer a unique platform for expanding the capabilities of natural biological systems for useful purposes, i.e. synthetic biology. They reduce complexity, remove structural barriers, and do not require the maintenance of cell viability. Cell‐free systems, however, have been limited by their inability to co‐activate multiple biochemical networks in a single integrated platform. Here, we report the assessment of biochemical reactions in an Escherichia coli cell‐free platform designed to activate natural metabolism, the Cytomim system. We reveal that central catabolism, oxidative phosphorylation, and protein synthesis can be co‐activated in a single reaction system. Never before have these complex systems been shown to be simultaneously activated without living cells. The Cytomim system therefore promises to provide the metabolic foundation for diverse ab initio cell‐free synthetic biology projects. In addition, we describe an improved Cytomim system with enhanced protein synthesis yields (up to 1200 mg/l in 2 h) and lower costs to facilitate production of protein therapeutics and biochemicals that are difficult to make in vivo because of their toxicity, complexity, or unusual cofactor requirements.     

Quantitative histochemical determination of succinic dehydrogenase activity in skeletal muscle fibres

Summary A histochemical technique was developed for the quantitative determination of succinic dehydrogenase (SDH) activity in muscle cross-sections using 1-methoxyphenazine methosulphate (mPMS) as the exogenous electron carrier, and azide as an inhibitor of cytochrome oxidase. The optimal composition of the incubation medium for the SDH reaction was determined. This histochemical procedure was compared to one using phenazine methosulphate (PMS) instead of mPMS and cyanide instead of azide. The substitution of mPMS and azide resulted in a substantial decrease in the non-specific reduction of nitroblue tetrazolium (NBT; the reaction indicator), i.e., nothing dehydrogenase activity. With mPMS and azide in the reaction medium, the production of NBT formazan was linear for at least 9 min during the enzymic reaction. This compared to a non-linear reduction of NBT during the initial stages of the reactions (SDH and nothing dehydrogenase) when using PMS and cyanide. The use of both mPMS and azide also eliminated the production of NBT monoformazan which occurred with PMS and cyanide. This procedure was shown to meet various criteria established for the quantification of histochemical reactions.     

Assessing evolutionary epistemology

There are two interrelated but distinct programs which go by the name evolutionary epistemology. One attempts to account for the characteristics of cognitive mechanisms in animals and humans by a straightforward extension of the biological theory of evolution to those aspects or traits of animals which are the biological substrates of cognitive activity, e.g., their brains, sensory systems, motor systems, etc. (EEM program). The other program attempts to account for the evaluation of ideas, scientific theories and culture in general by using models and metaphors drawn from evolutionary biology (EET program). The paper begins by distinguishing the two programs and discussing the relationship between them. The next section addresses the metaphorical and analogical relationship between evolutionary epistemology and evolutionary biology. Section IV treats the question of the locus of the epistemological problem in the light of an evolutionary analysis. The key questions here involve the relationship between evolutionary epistemology and traditional epistemology and the legitimacy of evolutionary epistemology as epistemology. Section V examines the underlying ontological presuppositions and implications of evolutionary epistemology. Finally, section VI, which is merely the sketch of a problem, addresses the parallel between evolutionary epistemology and evolutionary ethics.This research was supported, in part, by a grant from the National Science Foundation # SES—8308720. I want to thank Richard Lewontin and the Museum of Comparative Zoology, Harvard University for their hospitality and support during Spring 1984. I also want to thank David Hull, Charles Dyke and Michael Ruse for helpful comments on an earlier draft, and Andy Altman for pointing out to me the passage from Inherit the Wind.     

Individuals at the Center of Biology: Rudolf Leuckart’s <Emphasis Type="Italic">Polymorphismus der Individuen</Emphasis> and the Ongoing Narrative of Parts and Wholes. With an Annotated Translation

Rudolf Leuckart’s 1851 pamphlet Ueber den Polymorphismus der Individuen (On the polymorphism of individuals) stood at the heart of naturalists’ discussions on biological individuals, parts and wholes in mid-nineteenth-century Britain and Europe. Our analysis, which accompanies the first translation of this pamphlet into English, situates Leuckart’s contribution to these discussions in two ways. First, we present it as part of a complex conceptual knot involving not only individuality and the understanding of compound organisms, but also the alternation of generations, the division of labor in nature, and the possibility of finding general laws of the organic world. Leuckart’s pamphlet is important as a novel attempt to give order to the strands of this knot. It also solved a set of key biological problems in a way that avoided some of the drawbacks of an earlier teleological tradition. Second, we situate the pamphlet within a longer trajectory of inquiry into part-whole relations in biology from the mid-eighteenth century to the present. We argue that biological individuality, along with the problem-complexes with which it engaged, was as central a problem to naturalists before 1859 as evolution, and that Leuckart’s contributions to it left a long legacy that persisted well into the twentieth century. As biologists’ interests in part-whole relations are once again on the upswing, the longue durée of this problem merits renewed consideration.