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.     

Some algebraic properties of (M,R)-systems

On the basis of Rosen's representation of (M, R)-systems as sequential machines (Rosen,Bull. Math. Biophys.,26, 103–111, 1964), the existence of projective limits in categories of general (M, R)-systems is proved.     

Abstract biological systems as sequential machines

It is shown that a rather close relationship exists between the (ℳ,ℛ)-systems, defined previously as prototypes of abstract biological systems, and the sequential machines which have been studied by various authors. The theory of sequential machines is reformulated in a way suitable for its application to the study of the intertransformability of (ℳ,ℛ)-systems as a result of environmental alteration. The important concept of strong connectedness is most useful in this direction, and is used to derive a number of results on intertransformability. Some suggestions are made for further studies along these lines.     

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.     

A relational theory of biological systems II

The general Theory of Categories is applied to the study of the (M, R)-systems previously defined. A set of axioms is provided which characterize “abstract (M, R)-systems”, defined in terms of the Theory of Categories. It is shown that the replication of the repair components of these systems may be accounted for in a natural way within this framework, thereby obviating the need for anad hoc postulation of a replication mechanism. A time-lag structure is introduced into these abstract (M, R)-systems. In order to apply this structure to a discussion of the “morphology” of these systems, it is necessary to make certain assumptions which relate the morphology to the time lags. By so doing, a system of abstract biology is in effect constructed. In particular, a formulation of a general Principle of Optimal Design is proposed for these systems. It is shown under what conditions the repair mechanism of the system will be localized into a spherical region, suggestive of the nuclear arrangements in cells. The possibility of placing an abstract (M, R)-system into optimal form in more than one way is then investigated, and a necessary and sufficient condition for this occurrence is obtained. Some further implications of the above assumptions are then discussed.     

Sporadic sustained nonperiodic oscillations in the activities of organismic sets

In combining the author's theories of organismic sets (Rashevsky,Bull. Math. Biophysics,31, 159–198, 1969a) and Robert Rosen's theory of (M, R)-systems (Bull. Math. Biophysics,20, 245–265, 1958), a conclusion is reached that the number of either normal or pathological phenomena in organismic sets may occur. Those phenomena are characterized by occurring spontaneously once in a while but are not exactly periodic. Some epilepsies are an example of such pathological phenomena in the brain.     

Abstract biological systems as sequential machines: Behavioral reversibility

Rosen’s identification of abstract biological systems, called (M,R)-systems, with sequential machines is formally characterized. It is then shown that the determination of environmental alterations of (M,R)-systems from a knowledge of the response sequence and the structure of the system, which we call behavioral reversibility, can be interpreted as information-losslessness of sequential machines. Applying this relationship, necessary conditions for behavioral reversibility are derived. It is further shown that, similar to Rosen’s work on structural reversibility, (M,R)-systems are behaviorally reversible only if the number of physically realizable mappings are restricted.     

Abstract biological systems as sequential machines II: Strong connectedness and reversibility

It was previously shown that the abstract biological systems called. (ℳ, ℛ)-systems could be regarded formally as sequential machines, and that when this was done, the reversibility of environmentally induced structural changes in these systems was closely related to the strong connectedness of the corresponding machines. In the present work it is shown that the sequential machines arising in this way are characterized by the property that the size of the input alphabet is very small compared with the size of the set of states of the machine. It is further shown that machines with this property almost always fail to be strongly connected. Therefore, it follows that one of the following alternatives holds: either most environmentally induced structural alterations are not environmentally reversible, or else many mappings in the category from which the (ℳ, ℛ)-systems are formed must not be physically realizable.     

A note on replication in (M,R)-systems

The condition which allows the existence of induced replication maps in (M,R)-systems is shown to place strong restrictions on the “richness” of the category from which these systems can be constructed. This condition also admits of a simple biological interpretation, which can be checked empirically, and which may offer insight into the physical and biological realizations of these abstract systems.     

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.     

The reaction force and the transition region of a reaction

The reaction force F(R) and the position-dependent reaction force constant κF(R) are defined by F(R)=-∂V(R)/∂R and κ(R)=∂²V(R)/∂R², where V(R) is the potential energy of a reacting system along a coordinate R. The minima and maxima of F(R) provide a natural division of the process into several regions. Those in which F(R) is increasing are where the most dramatic changes in electronic properties take place, and where the system goes from activated reactants (at the force minimum) to activated products (at the force maximum). κ(R) is negative throughout such a region. We summarize evidence supporting the idea that a reaction should be viewed as going through a transition region rather than through a single point transition state. A similar conclusion has come out of transition state spectroscopy. We describe this region as a chemically-active, or electronically-intensive, stage of the reaction, while the ones that precede and follow it are structurally-intensive. Finally, we briefly address the time dependence of the reaction force and the reaction force constant.     

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.     

On the reversibility of environmentally induced alterations in abstract biological systems

The environmentally induced alterations in structure of (M, ℜ) which were described previously (R. Rosen,Bull. Math. Biophysics,23, 165–171, 1961) are examined from the standpoint of determining under what circumstances they can be reversed by further environmental interactions. For simplicity we consider only the case of (M, ℜ)-systems possessing one “metabolic” and one “genetic” component. In the case of environmentally induced alteration of the “metabolic” component alone, a necessary and sufficient condition is given for the reversibility of the alteration. In the case of alteration of the “genetic” component, the situation becomes more complex; several partial results are given, but a full analysis is not available at this time. Some possible biological implications of this analysis are discussed. This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract no. AF-49(638)-917 and Grant no. AF-AFOSR-9-63.     

Unique Carotenoids in the Terrestrial Cyanobacterium <Emphasis Type="Italic">Nostoc commune</Emphasis> NIES-24: 2-Hydroxymyxol 2′-Fucoside,Nostoxanthin and Canthaxanthin

Cyanobacteria produce some carotenoids. We identified the molecular structures, including the stereochemistry, of all the carotenoids in the terrestrial cyanobacterium, Nostoc commune NIES-24 (IAM M-13). The major carotenoid was β-carotene. Its hydroxyl derivatives were (3R)-β-cryptoxanthin, (3R,3′R)-zeaxanthin, (2R,3R,3′R)-caloxanthin, and (2R,3R,2′R,3′R)-nostoxanthin, and its keto derivatives were echinenone and canthaxanthin. The unique myxol glycosides were (3R,2′S)-myxol 2′-fucoside and (2R,3R,2′S)-2-hydroxymyxol 2′-fucoside. This is only the second species found to contain 2-hydroxymyxol. We propose possible carotenogenesis pathways based on our identification of the carotenoids: the hydroxyl pathway produced nostoxanthin via zeaxanthin from β-carotene, the keto pathway produced canthaxanthin from β-carotene, and the myxol pathway produced 2-hydroxymyxol 2′-fucoside via myxol 2′-fucoside. This cyanobacterium was found to contain many kinds of carotenoids and also displayed many carotenogenesis pathways, while other cyanobacteria lack some carotenoids and a part of carotenogenesis pathways compared with this cyanobacterium.     

Natural transformations of organismic structures

The mathematical structures underlying the theories of organismic sets, (M, R)-systems and molecular sets are shown to be transformed naturally within the theory of categories and functors. Their natural transformations allow the comparison of distinct entities, as well as the modelling of dynamics in “organismic” structures.     

Alcohols as Replacements of the Central Amide in β-Turns, Synthesis of Pro-Aib Hydroxyethylene Isostere and Analysis in Model β-Turn Peptides

Pro-Aib hydroxyethylene isosteres (S,R)- and (S,S)-7 were synthesized by cascade addition of 2-methyl-1-propenylmagnesium bromide to Boc-Pro-OMe in the presence of CuCN, followed by ketone reduction and olefin oxidation. By protecting the amine and hydroxyl groups in an oxazolidinone ring, hydroxyethylene isosteres 7 were successfully incorporated into Boc-Phe-Pro- ψ -[CH-(OH)-CH₂]-Aib-NHBn(α -Me) (S,R)-and (S,S)-11, which were characterized by ¹H NMR and IR spectroscopy. Examination of the NOESY spectra and the influence of solvent changes on the chemical shifts of the amide and carbamate proton signals for (S,R)-and (S,S)-11 indicated that both hydroxyethylene isosteres could adopt compact turn structures. The alcohol appears to act as a hydrogen donor in a seven-membered ring intramolecular hydrogen bond. In addition, analysis of the respective peptide (S,S)-16, in which the hydroxyl group was masked as a methyl ether, showed that the turn conformation was disrupted, and indicated the importance of the alcohol as a hydrogen-bond donor for turn stability. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

A note on abstract relational biologies

It is shown that the class of abstract block diagrams of (M, ℜ)-systems which can be constructed out of the objects and mappings of a particular subcategoryG ₀ of the categoryG of all sets depends heavily on the structure ofG ₀, and in particular on the number of sets of mappingsH(A, B) which are empty inG ₀. In the context ofG ₀-systems, there-fore, each particular categoryG ₀ gives rise to a different “abstract biology” in the sense of Rashevsky. A number of theorems illustrating the relation between the structure of a categoryG ₀ and the embeddability of an arbitrary mapping αεG ₀ into an (M, ℜ)-system are proved, and their biological implication is discussed. This research was supported by the United States Air Force through the Air Force Office of Scientific Reserch of the Air Research and Development Command, under Contract No. AF 49(638)-917.     

Cellular systems as sequential machines: Stability properties

In previous studies of (M,R) (Rosen, 1961; Demetrius, 1966), it was assumed that changes in the structure of (M,R) which were induced by environmental alternations occurred without error. Here, the effect of both “genetic” and “metabolic” malfunctions on the behavior of (M,R) is examined and a subclass of these systems whose behavior is invulnerable to such errors is specified.     

Automobile driving as psychophysical discrimination

The driver tries to keep the car in the center of the lane. If the car is too near the left edge, this causes the driver to make a “corrective” right turn. If the car is near the right edge, a “corrective” left turn is made. Therefore, a quantity which decreases with increasing distance Δ _L from the left edge may be considered as a stimulusS _R producing the reactionR _R of turning to the right. A similar situation holds for the distance Δ _R from the right edge. When the car is in the center of the lane, Δ _L = Δ _R andS _R =S _L , the stimuli are equal. We thus have here a situation analogous to the one studied by H. D. Landahl in his theory of psychophysical discrimination. In general a reactionR _R (resp.R _L ) will occur only ifR _R −R _L ≥h ^* (resp.R _L −R _R ≥h ^*) whereh ^* is a threshold. Applying Landahl’s theory to this situation, we find thath ^* determines the distance from the edge, at which a corrective turn is made. This distance is not constant, but a function of the speedv of the car. The requirement that a corrective turn should be madebeforre the car runs off the road leads to an expression for the maximum safe speed. Because of the transcendency of the equations involved, closed solutions cannot be obtained. It is, however, shown that the expression for maximum safe speed, given in a previous paper (Bull. Math. Biophysics,21, 299–308, 1959), is a rough first approximation to the expressions found now.