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.     

Some comments on re-estalishability

The present note consists of two separate but related parts. In the first, a new graphtheoretic proof is presented that an (ℳ,R)-system must always contain a nonreestablishable component. The second considers some questions concerning the relation between re-establishability and the time-lag structure in (ℳ,R)-systems. It is supposed that the reader is familiar with the terminology of the author's previous work on (ℳ,R)-systems, particularly R. Rosen,Bull. Math. Biophysics,20, 245–260, 1958.     

Categories of (ℓ, ℛ)-systems

We show that when we represent (ℓ, ℛ)-systems with fixed genome as automata (sequential machines), we get automata with output-dependent states. This yields a short proof that ((ℓ, ℛ)-systems from a subcategory of automata—and with more homomorphisms than previously exhibited. We show how ((ℓ, ℛ)-systems with variable genetic structure may be represented as automata and use this embedding to set up a larger subcategory of the category of automata. An analogy with dynamical systems is briefly discussed. This paper presents a formal exploration and extension of some of the ideas presented by Rosen (Bull. Math. Biophyss,26, 103–111, 1964;28, 141–148;28 149–151). We refer the reader to these papers, and references cited therein, for a discussion of the relevance of this material to relational biology.     

The duration of a sequence of epidemics

The typical duration of an epidemic in a sequence of linearly ordered populations shows a surprising nonmonotonic behaviour with respect to population size, which was noted by Swinton (1998) [Bull. Math. Biol., 60, 215–230]. This paper gives the sketch of a proof of the phenomenon.     

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.     

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.     

Contributions to relational biology

The principle of biotopological mapping (Rashevsky, 1954,Bull. Math. Biophysics,16, 317–48) is given a generalized formulation, as the principle of relational epimorphism in biology. The connection between this principle and Robert Rosen’s representation of organisms by means of categories (1958,Bull. Math. Biophysics,20, 317–41) is studied. Rosen’s theory of (M,R)-systems, (1958,Bull. Math. Biophysics,20, 245–60) is generalized by dropping the assumption that only terminalM _i components are sending inputs into theR _i components. It is shown that, if the primordial organism is an (M,R)-system, then the higher organisms, obtained by a construction well discussed previously (1958,Bull. Math. Biophysics,20, 71–93), are also (M,R)-systems. Several theorems about such derived (M,R)-systems are demonstrated. It is shown that Rosen’s concept of an organism as a set of mappings throws light on phenomena of synesthesia and also leads to the conclusion that Gestalt phenomena must occur not only in the fields of visual and auditory perception but in perceptions of any modality.     

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.     

A quantum-theoretic approach to enzyme specificity: Proposals for a new type of biochemical control mechanism

A quantum-theoretic approach to the problem of enzymic specificity is presented. The concept of a “measuring system” analogized with the enzyme is utilized. Along these lines a quantum mechanical hypothesis for the mechanism of enzyme reactions was advanced (Enzymologia,35, 117–130, 1968). In the measuring process an ℐ-observable, linked to the proper values of the substrate'sA ₁...A _m observables will indicate the state of the measuring apparatus. On the corresponding Hilbert space of the enzymesubstrate complexH _ES , through the respective statistical operatorU _ES we get a “state” vector [s, a]. Theng:Γ _S xΓ _E →Γ _ES , that is, to an oriented pair 〈s, a〉 ∈Γ _S xΓ _E will correspond a “state” vector [s, a], and to a proper valuei _k of ℐ will correspond, throughg, the respective equivalence classes of Г-spaces. Introducing the concept of enzyme-substrate complex entropyL _ES , a theorem is demonstrated asserting thatL _S =L _ES -L _E + 1/2kn wheren is the number of the degrees of freedom which may fluctuate. The values ofL _ES are denoted “specific values,” and it is demonstrated that a microphysical systemS may be a substrate specific forE, if and only if it can realize one of theL _ES specific values. Along these lines a model of a stochastic process for the enzymic reaction is constructed, and the set of Kolmogorov equations for the respective probabilities is derived. When a “perturbation” is induced in our model, an interesting prediction concerning the fluctuations in the kinetics of the corresponding enzyme is obtained. A relevant experimental proof ensued from these theoretical considerations. When a “gentle perturbation” was induced in a substrate by mild X-ray or UV-irradiations, an interesting oscillatory behavior of enzymic activity was recorded. A biochemical control mechanism is constructed (a simple “flip-flop” type) utilizing nonoverlapping oscillations in the activity of two enzymes at a key metabolic intermediate level. Eventually, a relevant experimental proof for the respective control model is presented.     

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.     

Polar steroids from <Emphasis Type="Italic">Solaster endeca</Emphasis> starfish and the physiological activity of polar steroids from three starfish species

Four polyhydroxylated steroids, new (20R)-5α-cholestan-3β,6α,8,15α,24,26-hexaol (I) and known (20R,25S)-5α-cholestan-3β,6α,8,15β,16β,26-hexaol, (20R,25S)-5α-cholestan-3β,6α,15β,16β,26-pentaol, and marthasterone sulfate were isolated from the Solaster endeca starfish inhabiting the Sea of Okhotsk and characterized. Steroid (I) contains a 24,26-dihydroxylated side chain, which is unique for starfish polyols. The isolated steroids and related metabolites from two starfish species of the Evasterias genus (in total, 15 compounds) were weakly cytotoxic in a human HeLa cell culture and some of them were inhibitors of non-specific esterase from mouse Ehrlich carcinoma. The effects of these compounds on the p53 protein activity were studied in a yeast two-hybrid test system and both inhibitors and stimulators of this activity were found among them.     

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.     

On the number of balanced signed graphs

The classical enumeration theorem of Pólya (Acta Math.,68, 145–254, 1937) is applied to a modified version of Harary’s (Pacific J. Math.,8, 743–755, 1958) generating functions for counting bicolored graphs to derive a counting function for the number of balanced signed graphs. Methods for computing these counting polynomial functions are discussed.     

Antifoulant diterpenes produced by the brown seaweed Canistrocarpus cervicornis

This paper reports the antifouling properties of the dichloromethane crude extract (DC) and 3 pure compounds isolated from the Brazilian brown seaweed Canistrocarpus cervicornis against establishment of the mussel Perna perna. DC extract showed a strong inhibition activity against byssal threads. Two natural dolastanes and one seco-dolastane diterpene, namely (4R, 9S, 14S)-4α-acetoxy-9β,14α-dihydroxydolast-1(15),7-diene (1), (4R,7R,14S)-4α,7α-diacetoxy-14-hydroxydolast-1(15),8-diene (2) and isolinearol (3), were isolated from DC extract. Dolastane (1) inhibited 60% of byssal fixation, while compound 2 and the seco-dolastane (3) strongly inhibited (82%) the establishment of P. perna. This is the first report of this type of chemical skeleton in three powerful compounds that could be further explored for the development of antifouling technology.     

Applications of the ETS-NOCV method in descriptions of chemical reactions

The present study characterizes changes in the electronic structure of reactants during chemical reactions based on the combined charge and energy decomposition scheme, ETS-NOCV (extended transition state–natural orbitals for chemical valence). Decomposition of the activation barrier, ΔE ^#, into stabilizing (orbital interaction, ΔE _orb, and electrostatic, ΔE _elstat) and destabilizing (Pauli repulsion, ΔE _Pauli, and geometry distortion energy, ΔE _dist) factors is discussed in detail for the following reactions: (I) hydrogen cyanide to hydrogen isocyanide, HCN → CNH isomerization; (II) Diels-Alder cycloaddition of ethene to 1,3-butadiene; and two catalytic processes, i.e., (III) insertion of ethylene into the metal-alkyl bond using half-titanocene with phenyl-phenoxy ligand catalyst; and (IV) B–H bond activation catalyzed by an Ir-containing catalyst. Various reference states for fragments were applied in ETS-NOCV analysis. We found that NOCV-based deformation densities (Δρ _i) and the corresponding energies ΔE _orb(i) obtained from the ETS-NOCV scheme provide a very useful picture, both qualitatively and quantitatively, of electronic density reorganization along the considered reaction pathways. Decomposition of the barrier ΔE^# into stabilizing and destabilizing contributions allowed us to conclude that the main factor responsible for the existence of positive values of ΔE ^# for all processes (I, II, III and IV) is Pauli interaction, which is the origin of steric repulsion. In addition, in the case of reactions II, III and IV, a significant degree of structural deformation of the reactants, as measured by the geometry distortion energy, plays an important role. Depending on the reaction type, stabilization of the transition state (relatively to the reactants) originating either from the orbital interaction term or from electrostatic attraction can be of vital importance. Finally, use of the ETS-NOCV method to describe catalytic reactions allows extraction of information on the role of catalysts in determination of ΔE ^#.     

Primitivity of products of leslie matrices

An earlier paper by E. L. Keller, 1980,Bull. math. Biol. 42, 181–189, gave necessary and sufficient conditions for primitivity of a product of two Leslie matrices. This paper generalized that result to an arbitrary number of Leslie matrices. The generalized result shows that Keller's is, from one viewpoint, more conveniently stated in a slightly different form (our equation (6)). This restatement in turn simplifies the proof of his special case of the general result. It also shows that the case of products of just pairs of matrices is a special one in that the simple form of result obtained for that case doesnot retain its simplicity when generalized to higher order products.     

New synonymies for <Emphasis Type="Italic">Ruellia</Emphasis> (Acanthaceae) of Costa Rica and notes on other neotropical species

As part of our study of the Acanthaceae of Costa Rica and from a phylogenetic study of Ruellia as a whole, we place R. barbillana and R. metallica in synonymy with R. terminalis. We discuss species limits of R. terminalis and provide a key that distinguishes it from close relatives including R. grantii, R. oaxacana, and R. phyllocalyx. Ruellia puri is put into synonymy with R. jussieuoides, and R. standleyi is synonymized with R. ochroleuca. Disjunct distributions characterize R. terminalis, R. jussieuoides, and R. ochroleuca, although this pattern is most extreme in R. ochroleuca. Phylogenetic analysis of DNA sequence data lends support to our synonymy decisions. New lectotypifications are provided for Arrhostoxylum achimeniflorum, Dipteracanthus puri, Dipteracanthus puri β angustifolius, Dipteracanthus puri γ gymnocladus, Lychniothyrsus ochroleucus, and Otacanthus pearcei .     

Regioselective deglycosylation of onion quercetin glucosides by <Emphasis Type="Italic">Saccharomyces cerevisiae</Emphasis>

Bioconversion of quercetin glucosides using four generally recognized as safe (GRAS) organisms (Aspergillus oryzae, Bacillus subtilis, Lactobacillus plantarum, and Saccharomyces cerevisiae) was evaluated by measuring changes in the levels of quercetin compounds of onion. Of the four organisms, S. cerevisiae increased the content of quercetin-3-O-β-d-glucoside (III; isoquercitrin) and quercetin (IV), whereas decreasing quercetin-3,4′-O-β-d-glucoside (I) and quercetin-4′-O-β-d-glucoside (II). Also, S. cerevisiae converted authentic compound I to III, and II to IV, respectively. These results suggest that S. cerevisiae can be used to increase the levels of isoquercitrin (III), the most bioavailable quercetin compound in onion.     

Bioactive metabolites from <Emphasis Type="Italic">Phoma</Emphasis> species,an endophytic fungus from the Chinese medicinal plant <Emphasis Type="Italic">Arisaema erubescens</Emphasis>

Through bioassay-guided fractionation, the EtOAc extract of a culture broth of the endophytic fungus Phoma species ZJWCF006 in Arisaema erubescens afforded a new α-tetralone derivative, (3S)-3,6,7-trihydroxy-α-tetralone (1), together with cercosporamide (2), β-sitosterol (3), and trichodermin (4). The structures of compounds were established on the basis of spectroscopic analyses. Compounds 1, 2, and 3 were obtained from Phoma species for the first time. Additionally, the compounds were subjected to bioactivity assays, including antimicrobial activity, against four plant pathogenic fungi (Fusarium oxysporium, Rhizoctonia solani, Colletotrichum gloeosporioides, and Magnaporthe oryzae) and two plant pathogenic bacteria (Xanthomonas campestris and Xanthomonas oryzae), as well as in vitro antitumor activities against HT-29, SMMC-772, MCF-7, HL-60, MGC80-3, and P388 cell lines. Compound 1 showed growth inhibition against F. oxysporium and R. solani with EC₅₀ values of 413.22 and 48.5 μg/mL, respectively. Additionally, compound 1 showed no cytotoxicity, whereas compound 2 exhibited cytotoxic activity against the six tumor cell lines tested, with IC₅₀ values of 9.3 ± 2.8, 27.87 ± 1.78, 48.79 ± 2.56, 37.57 ± 1.65, 27.83 ± 0.48, and 30.37 ± 0.28 μM, respectively. We conclude that endophytic Phoma are promising sources of natural bioactive and novel metabolites.     

A note on the development of organismic sets

It is suggested that the development of organismic sets is governed not by the maximalization of the integral survival value, as suggested previously (Bull. Math. Biophysics,28, 283–308, 1966;29, 139–152, 1967;30, 163–174, 1968), but by maximizing the number of new relations which appear as an organismic set develops.