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.     

Homeostatic control of thyroxin concentration expressed by a set of linear differential equations

The purpose of this work is to express current concepts on the relationship between the rates of secretion of thyroxin and of thyroid stimulating hormone (TSH) by a set of linear differential equations (two attempts have been made previously in this direction; cf. Roston,Bull. Math. Biophysics,21, 271–282, 1959; Danziger and Elmergreen,Bull. Math. Biophysics,16, 15–21, 1954), and to show that the solutions to these equations fulfill two criteria: that they correctly express the previously observed behavior of thyroxin and TSH, and that they allow certain predictions to be made which are amenable to experimental verification or disproval by currently existing techniques. This mathematical model is necessarily only an approximation of reality.     

On relations between sets: II

The notion of relations between sets, defined in a previous publication (Bull. Math. Biophysics,23, 233–235, 1961) is generalized and some biological examples are given. A generalization ton-ary relation is suggested.     

Kinetic study of DNA binding of cisplatin and of a bicycloalkyl-substituted (ethylenediamine)dichloroplatinum(II) complex

 Salmon sperm DNA platination has been conducted under strictly pseudo-first-order conditions with cisplatin (1) and rac-{(1S,2S,4S)-exo-2-(aminomethyl)-2-amino-7-bicyclo[2.2.1]heptane}dichloroplatinum(II) (2). An aquation step first occurs for both complexes, with the rate constants k ₁ = 1.12(0.02)×10^–4 s^–1 and 1.47(0.02)×10^–4 s^–1 respectively for 1 and 2 at 37  °C, values in agreement with those previously reported. It is followed by the actual platination step whose second-order rate constant has been determined for the first time by physicochemical techniques. The values for 1 and 2 respectively are: k ₂ = 2.08(0.07) M^–1 s^–1 and 3.9(0.4) M^–1 s^–1. These kinetic data are discussed in the context of a comparison of several biological properties of the two complexes. Received: 15 May 1998 / Accepted: 26 June 1998     

The representation of biological systems from the standpoint of the theory of categories

A mathematical framework for a rigorous theory of general systems is constructed, using the notions of the theory of Categories and Functors introduced by Eilenberg and MacLane (1945,Trans. Am. Math. Soc.,58, 231–94). A short discussion of the basic ideas is given, and their possible application to the theory of biological systems is discussed. On the basis of these considerations, a number of results are proved, including the possibility of selecting a unique representative (a “canonical form”) from a family of mathematical objects, all of which represent the same system. As an example, the representation of the neural net and the finite automaton is constructed in terms of our general theory.     

Kinetics of formation and stability of {Pt(dien)}2+ complexes with octamer and 14-mer DNA oligonucleotides containing a GG sequence

 Reaction of [Pt(dien)Cl]⁺ (1) with the 14-mer oligonucleotide 5′-d(ATACATGGTACATA) (I) gave rise to two major species which corresponded to the 5′-G and 3′-G platinated monofunctional adducts, and a minor amount of the bis-platinated adduct formed during the later stages of the reaction. The reaction of (1) with the related octamer 5′-d(ATACATGG) (II) was also investigated. Kinetic data obtained by HPLC showed that the 5′-G and 3′-G bases of the 14-mer oligonucleotide were platinated at similar rates: the second-order rate constant is 53×10^–2 M^–1 s^–1 at 298 K in 0.1 M NaClO₄. However, the platination rate of 5′-G of the octamer (II) (k=69×10^–2 M^–1 s^–1) was enhanced by a factor of three compared to the rate of platination at 3′-G (k=22×10^–2 M^–1 s^–1). All the adducts were separated by HPLC and characterized by NMR spectroscopy, enzymatic digestion and MALDI-TOF mass spectrometry. ¹H and ¹⁵N NMR shifts suggest that there are distinct conformational differences between 14-mer duplexes platinated at 5′-G (I5′ _ds) and 3′–G (I3′ _ds). Molecular mechanics modelling indicates that rotation around the Pt-N7 bond is more restricted in the case of the 5′-G adduct than in that of the 3′-G adduct. The binding of {Pt(dien)}²⁺ to 5′-GN7 and 3′-GN7 in the monofunctional adducts of (I) was shown to be reversible upon the addition of high concentrations of chloride ions. Received: 3 July 1998 / Accepted: 10 November 1998     

Two remarks on the mathematical biology of automobile driving

In connection with a series of previous papers by this author (Bulletin of Mathematical Biophysics,21, 299–308, 375–385;22, 257–262, 263–267;23, 19–29;24, 319–325) results obtained by A. Crawford (Economics 5, 417–428) on the effects of irrelevant lights on reaction times toward a given light stimulus are discussed. The conclusions from a previous paper of this author (Bulletin of Mathematical Biophysics,23, 19–29) are elaborated.     

A sociological approach to biology

The paper develops further some suggestions made previously (Bulletin of Mathematical Biophysics,28, 283–308, 1966) that certain biological phenomena may be more easily interpreted from a “sociological” point of view by considering the organism as a social aggregate of cells and a cell as a social aggregate of genes. In this light the problems of origin of life on earth, of aging, and of parasitism and symbiosis are discussed. The notion of social aggregates of different orders is introduced.     

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.     

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.     

On relations between sets: III

The theory of relations between sets, proposed and outlined in previous publications (Bull. Math. Biophysics,23, 233–235, 1961;28, 117–124, 1966;28, 309–313, 1966), is tentatively expanded and generalized with a view to biological applications.     

Further studies of some aspects of a neurobiophysical model of schizophrenia

The neurobiophysical model of schizophrenia discussed previously (Bull. Math. Biophysics,26, 167–185, 1964;27, 21–26, 1965) is generalized further, to include catatonic and stuporous states. It is concluded that the development of schizophrenia will proceed through different stages of catatonic and non-catatonic states, depending on parameters which characterize on one hand the general inhibition of the individual, on the other hand what may be called his “stability.” Suggestions for possible clinical verifications of the conclusions are made.     

A derivation of a rote learning curve from the total uncertainty of a task

The derivation of H. D. Landahl’s learning curve (1941,Bull. Math. Biophysics,3, 71–77) from a single information-theoretical assumption obtained previously (Rapoport, 1956,Bull. Math. Biophysics,18, 317–21) is extended to obtain the entire family of such curves with the number of stimuliM (to each of which one ofN responses is to be associated) as a parameter. No additional assumptions are required. The entire family thus appears as a function of a single free parameter,k, all other parameters being experimentally determined. The theory is compared with a set of experiments involving the learning of artificial languages. An alternative quasi-neurological model leading to the same equation is offered.     

Mechanism of intermolecular hydroacylation of vinylsilanes catalyzed by a rhodium(I) olefin complex: a DFT study

Density functional theory (DFT) was used to investigate the Rh(I)-catalyzed intermolecular hydroacylation of vinylsilane with benzaldehyde. All intermediates and transition states were optimized completely at the B3LYP/6-31G(d,p) level (LANL2DZ(f) for Rh). Calculations indicated that Rh(I)-catalyzed intermolecular hydroacylation is exergonic, and the total free energy released is −110 kJ mol⁻¹. Rh(I)-catalyzed intermolecular hydroacylation mainly involves the active catalyst CA2, rhodium–alkene–benzaldehyde complex M1, rhodium–alkene–hydrogen–acyl complex M2, rhodium–alkyl–acyl complex M3, rhodium–alkyl–carbonyl–phenyl complex M4, rhodium–acyl–phenyl complex M5, and rhodium–ketone complex M6. The reaction pathway CA2 + R2 → M1b → T1b → M2b → T2b1 → M3b1 → T4b → M4b → T5b → M5b → T6b → M6b → P2 is the most favorable among all reaction channels of Rh(I)-catalyzed intermolecular hydroacylation. The reductive elimination reaction is the rate-determining step for this pathway, and the dominant product predicted theoretically is the linear ketone, which is consistent with Brookhart’s experiments. Solvation has a significant effect, and it greatly decreases the free energies of all species. The use of the ligand Cp′ (Cp′ = C₅Me₄CF₃) decreased the free energies in general, and in this case the rate-determining step was again the reductive elimination reaction.     

Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma

The purpose of this paper is to present a mathematical model for the tumor vascularization theory of tumor growth proposed by Judah Folkman in the early 1970s and subsequently established experimentally by him and his coworkers [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 53–65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323–328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: an experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413–419]. In the simplest version of this model, an avascular tumor secretes a tumor growth factor (TGF) which is transported across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the TGF gradient back to the tumor, proliferating and forming a new capillary network. In the model presented here, we include two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as a protease inhibitor. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a subcase. Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks; (3) standard transport equations for the diffusion of molecular species in porous media. One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 73–65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323–328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: An experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413–419]. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates as it approaches the tumor [Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64]. An erratum to this article is available at .     

Some further comments on the DNA-protein coding problem

A notion of quasi-ergodicity is defined in free monoids, generalizing a notion introduced in a previous paper (Rosen, 1959,Bull. Math. Biophysics,21, 71–95). It is shown that under certain conditions the algebraic properties of quasi-ergodicity are very similar to those derived for the more specialized concept inloc. cit. If it is assumed that the DNA-protein coding processes in nature are of a quasi-ergodic nature, then a condition is specified under which only a finite number of different DNA-protein codes are possible, and an upper bound for the total number of different quasi-ergodic codes is obtained.     

Photodimerizations of hydroxy- and benzoylated 4-azachalcones and quantum chemical investigation of the reactions

Photodimerization reactions of compounds 4–6 gave four new cyclobutane-containing compounds (7–9) with full control over the stereochemistry at the four stereogenic centers. These new cyclobutane-containing compounds had β-truxinic (7a), δ-truxinic (7b and 9), and ε-truxillic (8) structures. However, o-, m-, and p-hydroxy 4-azachalcones (1–3) did not give photochemical cyclization products under any conditions (in solvent or in their solid or molten states). Experimental data suggested the possibility of frontier orbital control over stereochemical behavior, so some theoretical calculations were performed. Full geometrical optimization of compounds 1–9 was performed via DFT B3LYP/6-31⁺G**, and their electronic structures were also investigated. The geometries of the singlet and triplet states were initially optimized by density functional theory (DFT) and the configuration interaction singles (CIS) B3LYP/3-21⁺G** level. An additional calculation was performed for the triplet state using the ground-state geometry. The possible photochemical dimerization products of compounds 7–9 (a–g) and the intrinsic reaction coordinates (IRCs) of the reactions of compounds 4–6 were calculated theoretically by the DFT/3-21⁺G** method. The configurations (reactant, transition state, product, and reaction pathway) corresponding to the stationary points (minima or saddle points) were determined. The intrinsic reaction coordinates were followed to verify the energy profiles that connect each TS to the appropriate local minimum. The dimeric products expected from the calculations coincided with the dimers produced experimentally.     

Analysis of tracer experiments: IV. The kinetics of generalN compartment systems

The methods of C. W. Sheppard and A. S. Householder (Jour. App. Physcis,22, 510–20, 1951), H. D. Landahl (Bull. Math. Biophysics,16, 151–54, 1954) and H. E. Hart (Bull. Math. Biophysics,17, 87–94, 1955;ibid.,19, 61–72, 1957;ibid.,20, 281–87, 1958) are employed in studying the kinetics of generalN compartment systems. It is shown that the nature of the transfer processes occurring in fluid flow systems and the chemical processes occurring in quadratic systems and in catalyzed quadratic systems can in principle be completely determined for all polynomial dependencies. Systems involving three-body and higher-order interactions can be completely solved, however, only if supplementary information is available. Research supported by the Atomic Energy Commission, Contract AT (30-1)-1551.     

Studies in the Capparaceae XXIII: Capparis coimbrana, a new species from Bolivia

Capparis coimbrana, a tree of the Bolivian and adjoining Brazilian subtropical to warm-temperate seasonal dry forests and savannas, is segregated fromC. petiolaris of the Pacific slope of Ecuador and northern Peru by its subcylindrical fruits (vs. spherical), white pulp (vs. yellow), sepals 3–5 mm (vs.1–3 mm), leaves with 13–24 (vs. 8–15) lateral veins per side, and by 1000 km of geographic isolation. The new species has edible fruits, but is not domesticated. The specific epithet celebrates Germán Coimbra S., eminent Bolivian ethnobotanist.

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Resumen   Capparis coimbrana es un árbol del bosque seco y sabanas subtropicales y cálidas de Bolivia y la parte adyacente de Brasil, segregada deCapparis petiolaris, que se encuentra distribuida en la región del Pacífico de Ecuador y norte de Perú, por sus frutos subcilíndricos (vs. esféricos), pulpa blanca (vs. amarilla), sépalos 3–5 mm (vs. 1–3 mm), hojas con 13–24 (vs. 8–15) pares de venas laterales, y aislada geográficamente 1000 km. Esta nueva especie posee frutos comestibles, pero no está domesticada. Su cpíteto específico honra a Germán Coimbra S., eminente etnobotánico boliviano.