Two definitions of stability of equilibria

In classical physics the stability of an equilibrium requires that any, even infinitesimal, displacement from the configuration of equilibrium results in forces which tend to restore the original equilibrium configuration. In case of several stable equilibrium configurations, the height of the threshold, which must be exceeded by the deviarion from the stable equilibrium in order to bring the configuration into another stable equilibrium is taken as a measure of stability of the first configuration. In quantum mechanics, and in the recent work of I. Baianu, S. Comorosan and M. Marinescu (Bull. Math. Biophysics,30, 625–635, 1968;31, 59–70, 1969;32, 539–561, 1970) on organismic supercategories, preference is given to take, as ameasure of the degree of stability of a configuration, or of a “state”, the length of time during which the system remains in that configuration. It is shown that under rather general conditions the two criteria are equivalent.     

On the fisher and the cubic-polynomial equations for the propagation of species properties

For precise boundary conditions of biological relevance, it is proved that the steadily propagating plane-wave solution to the Fisher equation requires the unique (eigenvalue) velocity of advance 2(Df)^1/2, whereD is the diffusivity of the mutant species andf is the frequency of selection in favor of the mutant. This rigorous result shows that a so-called “wrong equation”, i.e. one which differs from Fisher's by a term that is seemingly inconsequential for certain initial conditions, cannot be employed readily to obtain approximate solutions to Fisher's, for the two equations will often have qualitatively different manifolds of exact solutions. It is noted that the Fisher equation itself may be inappropriate in certain biological contexts owing to the manifest instability of the lowerconcentration uniform equilibrium state (UES). Depicting the persistence of a mutantdeficient spatial pocket, an exact steady-state solution to the Fisher equation is presented. As an alternative and perhaps more faithful model equation for the propagation of certain species properties through a homogeneous population, we consider a reaction-diffusion equation that features a cubic-polynomial rate expression in the species concentration, with two stable UES and one intermediate unstable UES. This equation admits a remarkably simple exact analytical solution to the steadily propagating plane-wave eigenvalue problem. In the latter solution, the sign of the eigenvelocity is such that the wave propagates to yield the “preferred” stable UES (namely, the one further removed from the unstable intermediate UES) at all spatial points ast→∞. The cubic-polynomial equation also admits an exact steady-state solution for a mutant-deficient or mutant-isolated spatial pocket. Finally, the perpetuating growth of a mutant population from an arbitrary localized initial distribution, a mathematical problem analogous to that for ignition in laminar flame theory, is studied by applying differential inequality analysis, and rigorous sufficient conditions for extinction are derived here.     

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 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.     

Parasitism and symbiosis in an N-person non-constant-sum continuous game

A non-constant-sum continuous game analyzed in a previous paper by one of the authors (A. Rapoport,Bull. Math. Biophysics,18, 15–30, 1956) is extended from two toN players with special emphasis onN=3. It is shown that the concept of stability becomes in this case one of “pairwise stability,” and depends on the so-called distribution matrix of rewards. The distribution matrix and the collusion structure jointly determine the end states of the game. Conditions which lead to the emergence of one, two, or no “parasites” are derived. An apparatus is described which provides a physical analogue of the game, making possible the isolation of behavioral variables under the prescribed conditions of the game.     

Models for reserpine effects on normal and brain-operated animals on a continuous reinforcement bar pressing schedule

Mathematical models are presented for the relationship between changes in the bar pressing rate of albino rats on a continuous reinforcement schedule and alterations of the brain amines norepinephrine and serotonin induced by electrolytic lesions in the hypothalamus and by the drug, reserpine. The discussion utilizes central concepts from some of the more prominent hypotheses on the relation of the amines to behavior and for the potentiation effects of the drug by the lesion (A. Carlsson, 1964. “Functional Significance of Drug-induced Changes in Brain Monoamine Levels.” InBiogenic Amines, eds. H. E. Himwich and W. A. Himwich, New York: Elsevier, 9–27; B. B. Brodie and E. Costa, 1962. “Some Current Views on Brain Monoamines.”Psychopharm. Service Centre Bull.,2, 1–25.) Laboratory data (F. Grabarits and J. Harvey, 1966. “The Effects of Reserpine on Behavior and on Brain Concentrations of Serotonin and Norepinephrine in Control Rats and Rats with Hypothalamic Lesions.”J. Pharm.,153, 403–411) are interpreted, and suggestions for further experimental and theoretical work are made.     

Role of secondary metabolites as defense chemicals against ice-ice disease bacteria in biofouler at carrageenophyte farms

Carrageenophyte farming is an expanding economical activity in North Borneo Island, Malaysia. During routine monitoring of “ice-ice” disease and epiphyte outbreak at commercial farms, it was apparent that culture lines were heavily (60–80%) infested with biofoulers, particularly Acanthophora spp. and Laurencia majuscula. However, only L. majuscula showed dominance and flourished even during “ice-ice” disease outbreak. Presence of chemical defense against seaweed pathogens was investigated in two populations of L. majuscula collected from three major carrageenophyte farms in two districts; (A) Lohok Butun, Selakan Island, and Bum-Bum Island, in Semporna district, and (B) Telutuh, Carrington Reef, and Balambangan Island, in Kudat district. The first population contained elatol (1), and iso-obtusol (2), and, second population contained (Z)-10,15-dibromo-9-hydroxy-chamigra-1,3(15),7(14)-triene (3) and (E)-10-15-dibromo-9-hydroxy-chamigra-1,3(15),7(14)-triene (4), as their antibacterial metabolites. All four metabolites showed highly selective inhibition against “ice-ice” disease bacteria compared to human pathogens at 30 μg disk⁻¹. In addition, seasonal variation of these compounds at two representative farms (Selakan Island [P-1] and Balambangan Island [P-2]) revealed a 120–170% increase in concentration during “ice-ice” disease outbreak. Microscopy of fresh specimens showed the presence of corps en cerise, which is the synthesis and storage site of halogenated metabolites at superficial cortical cells, branch tips, and trichoblasts. This suggests the importance of these metabolites as defense chemicals against “ice-ice” disease bacteria in L. majuscula that grows on seaweed culture lines.     

A graphical method for analysing the dynamics of three-species systems

We report a series of experiments with three competing species and a novel graphical analysis that explores the dynamics of this simple multi-species system. The three competing species (Drosophila hydei, D. immigrans and D. virilis) were maintained in five very large cages, on a natural fruit resource. The analytic method involved constructing a “dynamic surface”, by interpolation, from the population trajectories. Within the dynamic surface constructed from the five experiments a small “equilibrium area” (an area of point vectors) could be identified. The implications of both the method and the results are discussed. Received: 22 July 1996 / Accepted: 29 April 1997     

Some remarks on the mathematical bio-sociology of the relativity of injustice

The author's theory of the adoption of certain types of behavior patterns (Rashevsky, N., 1957, “Contributions to the Theory Initiative Behavior”.Bull. Maths. Biophysics,19, 91–119; 1968,Looking at History through Mathematics, Cambridge, Massachusetts: M.I.T. Press) consisting of elementary behaviors for each of which there is an opposite one and the two are mutually exclusive, is applied to describe the changes in the general type of behavior of a society. The elementary acts of which the whole problem consists may be either overt activities or beliefs or opinions. The general behavior patternsadopted by the society are considered as the “proper” or “just” ones. Any deviation from it in either one or more of the component elementary behaviors is considered as “unjust” and is subject to some punitive action. The total number of possible mutually exclusive behavior patterns is very large but finite. Within this very large range of possible patterns, we find that this notion of justice is relative, because changes from any behavior pattern to any other may occur. It is further shown that the amount of punishment for the deviation from the accepted pattern in order to be effective as well as efficient must be applied in different ways to different individuals even for the same transgression.     

Chironomidae (Diptera) larvae assemblages differ along an altitudinal gradient and temporal periods in a subtropical montane stream in Northwest Argentina

The objective of this study was to analyze the spatial and temporal dynamics of Chironomidae larvae assemblages in a subtropical mountain river basin of northwestern Argentina. We hypothesized that they would respond to (1) environmental changes along the altitudinal gradient in the Yungas forests; (2) environmental changes caused by spates (e.g., increased discharge or sediment transportation). We sampled five sites along an altitudinal gradient of ca. 1,500 m (from 680 to 2,170 m) during low-water and high-water period for 2 years. The Chironomidae larvae assemblages varied along the altitudinal gradient and between sampling periods based on an indicator species analyses and an ordination analyses (NMDS). The best indicator for high altitude sites was “Genus X” followed by Onconeura sp.2, “Genus 9”, “Genus 10”, and Rheotanytarsus sp.2; while Rheotanytarsus lamellatus, Oliveiriella sanjavieri, and Thienemanniella sp. were good indicators for low altitude sites. O. sanjavieri and R. lamellatus were the best indicators for high-water period. The most relevant environmental variables influencing their spatial and temporal distribution were altitude, water temperature, conductivity, and pH. This study is the first to analyze and describe in detail the spatial and temporal distribution of Chironomidae larvae assemblages in northwestern Argentina, and most of the taxa reported here are new records for the region.     

In vitro separation of a rose chimera

“Fairmount 1 thorny” (“FM1 thorny”) (a Rosa multiflora Thunb ex. J. Murr.) and a thornless sport of “FM1 thorny” (“Fairmount 1” (“FM1”)) were established in vitro to investigate chimeral segregation under various levels of BA and to obtain a pure thornless rose. While the chimeral thornless sport was expected to segregate in vitro and yield both thorny and thornless plantlets, “FM1 thorny” was to yield only thorny plants. “FM1” segregated in vitro into its constituent genotypes and yielded thorny and thornless plantlets, suggesting that “FM1” is chimeral. “FM1 thorny” produced only thorny plants in vitro. These results indicate that the “FM1 thorny” clone was not chimeral (pure thorny) and that the thornless regenerates of “FM1” did not develop via somaclonal variation. There was a significant linear relationship between increasing BA concentration and the percentage of thorny plants. Among a population of 690 tissue culture derived plants from all the BA experiments, 6 plants were classified as pure thornless plants 1 year later.     

Sp1 polymorphism in collagen I α1 gene is associated with osteoporosis in lumbar spine of Mexican women

The Sp1 binding site polymorphism in collagen type I alpha 1 gene (COLIA1) has been associated with osteoporosis (OP) and bone mineral density (BMD). The aim of this study was to explore the association of this polymorphism with OP and BMD in the Mexican population by polymerase chain reaction and restriction fragment length polymorphism (PCR–RFLP) procedure. Allelic and genotypic frequencies from the Sp1 polymorphism were determined in 100 women with OP, 100 women without OP and 500 subjects from general Mexican population (GMP). Distribution of Sp1 polymorphism was in Hardy–Weinberg equilibrium. In spite of population structure due to racial mix in Mexican population, associations with OP were demonstrated. The frequency of “s” allele was significantly higher in women with OP (35%) than in women without OP (11%; P < 0.00001). Interestingly, “ss” genotype, was exclusive of women with OP and was associated with low BMD (0.588 ± 0.077 g/cm²) in contrast to “SS” genotype (0.733 ± 0.039 g/cm²; P = 0.0001). This work confirms the association of Sp1 polymorphism with low BMD and OP in Mexican population and make sure to use Sp1 as a genetic marker for OP in our population.     

Salt tolerance in Lycopersicon species VI. Genotype-by-salinity interaction in quantitative trait loci detection: constitutive and response QTLs

 A study of genotype-by-salinity interaction was carried out to compare the behavior of quantitative trait loci (QTLs) in two F₂ populations derived from crosses between the cherry tomato, Lycopersicon esculentum Mill. var. cerasiforme, and two wild relatives Lycopersicon pimpinellifolium (Jusl.) Mill. and Lycopersicon chesmannii f. minor (Hook. f.) Mull., grown at two environmental conditions (optimum and high salinity). QTLs for earliness and fruit yield could be classified into four groups: “response-sensitive”, those detected only under control conditions or whose contribution significantly decreased in salinity; “response-tolerant”, detected only in salinity or in which the direction of their additive effects changed; “constitutive”, detected in both growing conditions; and “altered” QTLs, those where the degree of dominance changed according to the presence or absence of salt. Epistatic interactions were also influenced by the salt treatment. This differential allele effect at some (non-constitutive) QTLs induced by salt stress will make selection under an “optimum environment” unfruitful for the “response-tolerant” QTLs. Similarly, selection under salinity will ignore “response-sensitive” QTLs. Given that salinity is highly variable in the field, marker-assisted selection should take into account not only the “response-tolerant” but also the “response-sensitive” QTLs although there might be cases where selection in some QTLs for both conditions is not feasible. Comparing both populations, very few QTLs showed the same behavior. Received: 5 August 1996 / Accepted: 25 October 1996     

Salt tolerance in Lycopersicon spp. VII. Pleiotropic action of genes controlling earliness on fruit yield

 The change from vegetative to reproductive development (earliness) in Lycopersicon chesmannii line L2 was delayed for 20 weeks when compared to other Lycopersicon species under greenhouse conditions. The interspecific hybrid of L. chesmannii L2 and L. esculentum E9, a cherry tomato cultivar, also showed this delay in reproductive development. The distribution of this character in the F₂-derived population showed a bimodal shape, plants could be scored easily as “early” or “late” in two nutrient conditions (optimum and high salinity). A QTL with major effects on earliness was detected in salinity, which explained 35.6% of the phenotypic variation. The effect of this QTL greatly diminished under control conditions, indicating differences in the genetic control of earliness between treatments. ACC synthase or phytochrome B2 are the products of candidate genes for such a major QTL. Other QTLs with minor effects, and epistatic interactions, are also involved in earliness under both conditions. A “late” F₂ subpopulation yielded twice as much as an “early” F₂; conversely, “early” plants were taller than “late” plants, regardless of the treatment. QTL analysis, carried out in both subpopulations, showed that yield differences may be explained by chesmannii alleles showing negative additive effects at some QTLs only in the “early” subpopulation. The effect of population subdivision on QTL analysis was investigated by computer simulations to show sample-size or random effects; thus, important pleiotropic or regulatory effects of genes controlling earliness on yield that affect QTL analysis, have been reveiled. Therefore alleles controlling earliness in L. chesmannii have to be taken into account for a more efficient utilization of the genetic resources of this species. Received: 30 June 1998 / Accepted: 31 August 1998     

“Chelatable iron pool”: inositol 1,2,3-trisphosphate fulfils the conditions required to be a safe cellular iron ligand

Mammalian cells contain a pool of iron that is not strongly bound to proteins, which can be detected with fluorescent chelating probes. The cellular ligands of this biologically important “chelatable”, “labile” or “transit” iron are not known. Proposed ligands are problematic, because they are saturated by magnesium under cellular conditions and/or because they are not “safe”, i.e. they allow iron to catalyse hydroxyl radical formation. Among small cellular molecules, certain inositol phosphates (InsPs) excel at complexing Fe³⁺ in such a “safe” manner in vitro. However, we previously calculated that the most abundant InsP, inositol hexakisphosphate, cannot interact with Fe³⁺ in the presence of cellular concentrations of Mg²⁺. In this work, we study the metal complexation behaviour of inositol 1,2,3-trisphosphate [Ins(1,2,3)P ₃], a cellular constituent of unknown function and the simplest InsP to display high-affinity, “safe”, iron complexation. We report thermodynamic constants for the interaction of Ins(1,2,3)P ₃ with Na⁺, K⁺, Mg²⁺, Ca²⁺, Cu²⁺, Fe²⁺ and Fe³⁺. Our calculations indicate that Ins(1,2,3)P ₃ can be expected to complex all available Fe³⁺ in a quantitative, 1:1 reaction, both in cytosol/nucleus and in acidic compartments, in which an important labile iron subpool is thought to exist. In addition, we calculate that the fluorescent iron probe calcein would strip Fe³⁺ from Ins(1,2,3)P ₃ under cellular conditions, and hence labile iron detected using this probe may include iron bound to Ins(1,2,3)P ₃. Therefore Ins(1,2,3)P ₃ is the first viable proposal for a transit iron ligand. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Phase change-related variations of dome shape in Eucalyptus urophylla × Eucalyptus grandis shoot apical meristems

Shoot apical meristem (SAM) domes derived from five different outdoor and in vitro sources of juvenile and mature Eucalyptus urophylla × Eucalyptus grandis akin genotypes were compared. Overall measurements of SAM dome height H and diameter D ranged from 2 to 35 μm and 20 to 80 μm, with significant differences according to the various physiological origins of plant material investigated. SAM domes from the mature trees “Mat” were taller than those from the rejuvenated ministock plants “Rej”; from the in vitro microcuttings “IVM” of the same clone and also from the in vitro juvenile seedlings “IVJ”, whereas outdoor seedlings “Juv” exhibited intermediate SAM dome height. SAM domes from the rejuvenated material “Rej”, from the in vitro mature “IVM” and juvenile “IVJ” origins were also narrower than those from the outdoor seedlings “Juv” and to lesser extent than those from the mature trees “Mat”. Overall, the mature source “Mat” displayed bigger and somehow sharper hemispherical domes than those from “Rej” and “Juv”, physiologically more juvenile, or those from the in vitro origins “IVM” and “IVJ” which looked flatter and smaller. SAM dome height, diameter D and H/D values varied also significantly according to the plastochron. More specifically, H, D, and H/D SAM differences between the five origins were not significant during the early plastochron phase corresponding to leaf initiation, to become more salient as leaf structures started to elongate and to differentiate. This was particularly obvious for mature tree “Mat” SAM dome shapes which showed at this stage much higher H/D values than the other SAM sources. A shape index S used for characterizing more accurately dome shape confirmed these trends. These observations provide additional arguments to the view that juvenility in trees becomes more and more time- and shoot-tip restricted as ageing increases in the course of time during the ontogenetical process and could be ultimately confined to the most organogenic phase of SAM, from which shoot characteristics derive.     

Demography of a dormancy-prone geophyte: influence of spatial scale on interpretation of dynamics

Both temporal and spatial scales are important in the evaluation of population dynamics, but the latter often receives less attention in demographic analyses. We used a 5-year demographic data set for a long-lived geophyte, Calochortus lyallii, to investigate the pattern and components of spatial variation at two scales (population and microsite). We found that neither the projected population structure nor asymptotic population growth rate (λ) varied greatly across either scale, although the underlying contributors to the variation in λ, V(λ), did differ between scales. Life table response experiment analyses showed that V(λ) among populations came primarily from variation in seedling survival and progression of non-reproductive plants, whereas V(λ) among microsites was primarily due to the variable fertility of large adults. Prolonged dormancy was important in reducing V(λ) among quadrats at both the scales, partly countering fluctuations in other transitions such as recruitment. This result represents some of the first evidence that underground “bulb banks” could function to offset the effects of a spatially heterogeneous environment in a manner analogous to seed banks. Future work is needed to isolate the specific, sometimes idiosyncratic, life history phenomena acting to modulate plant population dynamics in a spatial context.     

Spread of information through a population with socio-structural bias: I. Assumption of transitivity

A previously derived iteration formula for a random net was applied to some data on the spread of information through a population. It was found that if the axon density (the only free parameter in the formula) is determined by the first pair of experimental values, the predicted spread is much more rapid than the observed one. If the successive values of the “apparent axon density” are calculated from the successive experimental values, it is noticed that this quantity at first suffers a sharp drop from an initial high value to its lowest value and then gradually “recovers”. An attempt is made to account for this behavior of the apparent axon density in terms of the “assumption of transitivity”, based on a certain socio-structural bias, namely, that the likely contacts of two individuals who themselves have been in contact are expected to be strongly overlapping. The assumption of transitivity leads to a drop in the apparent axon density from an arbitrary initial value to the vicinity of unity (if the actual axon density is not too small). However, the “recovery” is not accounted for, and thus the predicted spread turns out to beslower than the observed.     

Pollination biology of <Emphasis Type="Italic">Disanthus cercidifolius</Emphasis> var. <Emphasis Type="Italic">longipes</Emphasis>, an endemic and endangered plant in China

Disanthus cercidifolius Maxim. var. longipes H.T. Chang usually has two inflorescences growing in opposite directions in the axillae, but occasionally three inflorescences grow paratactically. The typical flowering process could be divided into 4 periods: “Pre-dehiscence”, “Initial dehiscence”, “Full dehiscence” and “Withering”. Both the natural population and the planted population had a flowering peak of 15–35 days after the first flower bloomed. There were significant differences between the time courses of flowering of the two populations. Out-crossing is the main breeding system in this species. And autogamy decreases the risk of reproductive failure of this species. The main insect pollinators of D. cercidifolius var. longipes are Episyrphus balteatus de Geer, Scaptodrosophila coracina Kikkawa and Peng, Polistes olivaceus de Geer, Apis cerana Fabricius, Nezara viridula L. and Coccinella septempunctata L., and so on. Among the insects, S. coracina and E. balteatus are the most important and efficient pollinators, but others are inefficient pollinators. Though wind pollination is not efficient, it guarantees reproduction when insect pollinators are not available. “Mass flowering” is an adaptive behavior and reproductive strategy of this species, and “few fruiting” could be caused by the lack of pollinators.     

Abstract mathematical molecular biology: II some relational consequences of the “one gene-one enzyme” hypothesis

Following the program outlined in a previous paper (Bull. Math. Biophysics,23, 237–260, 1961), a further abstract study is made of some simple relational systems which possess some properties of living organisms. It is shown that the “one gene-one enzyme” hypothesis leads to the conclusion that either all genes are built of the same chemical building blocks, or that at least all genes have a number of building blocks in common. A consistent relational application of the “one gene-one enzyme” hypothesis leads moreover to the conclusion that replication is not an inherent property of a gene. Rather there must be a set of enzymes which “copy” the genes. The number of enzymes in this set must be less than the number of genes and therefore the activity of those “copying” enzymes cannot be absolutely specific.