On the Fokker-Planck equation in the stochastic theory of mortality: II

This is the continuation of part I, which was published in the September, 1963, issue ofThe Bulletin. Section 5 treats the special case in which the left absorbing barrier recedes to −∞, leaving essentially only one barrier at a finite distance Λ (>0) from the origin. The eigenfunctions are now parabolic cylinder functions. The limiting cases Λ→+∞ and Λ→0 are also considered. Though meaningless for practical applications to our problem, they are of interest, mathematically, because the Green’s function for the solution of the Fokker-Planck equation assumes a particularly simple form. In section 6 we study, by means of an example, how the “force of mortality” may vary with time before attaining its final asymptotic value. Section7, still dealing with only one absorbing barrier, shows that our results for “strong homeostasis” are identical with those derived by Chandrasekhar for the escape of particles through a potential barrier in the limiting case of quasi-static flow. Precise conditions are given for the validity of both the quasi-static and the Smoluchowski approximations to the Fokker-Planck equation. Finally, in section 8, a brief mention is made of Gevrey’s method for the solution of parabolic partial differential equations.     

On the mathematical theory of rumor spread

The applicability of the theory of random nets to the theory of rumor spread is shown. In particular the “weak connectivity” of the net appears as the saturation fraction of “knowers” in a thoroughly mixed population through which a message diffuses where each knower tells the message to a finite average number of individuals. Further it is shown how the time course equation of rumor spread, where time is measured by the number of “removes” from the starters, can be translated into an ordinary continuous time course equation if the distribution of the telling intervals is known.     

Molecular cloning and expression of α2,8-sialyltransferase (ST8Sia I, GD3 Synthase) in Xenopus

GD3, a minor ganglioside in most normal tissues, is involved in important biological events and its expression could increase in pathological conditions. Organism integrity requires a tight balance between the anabolic and catabolic processes, thus it is important to control the intracellular expression of those “key” enzymes, which act at the “branching point” of ganglioside metabolism; one of these is the GD3-synthase (ST8Sia I). In this paper, we report the sequences of two ST8Sia I mRNAs found in Xenopus laevis and their genomic organization; the canonical form resulted constituted of 5 exons and 4 introns, while the “short” mRNA lacks of the exon 2. The expression of the two ST8Sia I mRNAs during embryo development and their tissue distribution in adult animals showed the single or simultaneous presence of the two forms. Experiments of in vitro expression and evaluation of enzymatic activity of the two hypothetical proteins turned out to be ST8Sia I. In the end, considering the growing interest toward the specie Xenopus tropicalis, due to its diploid genome that render it more suitable for genetic studies, we also cloned X. tropicalis ST8Sia I. Accession numbers: AY272057, AY272056     

A statistical approach to the theory of the central nervous system

A “probabilistic” rather than a “deterministic” approach to the theory of neural nets is developed. Neural nets are characterized by certain parameters which give the probability distributions of different kinds of synaptic connections throughout the net. Given a “state” of the net (i.e., the distribution of firing neurons) at a given moment, an equation for the state at the next moment of quantized time is deduced. Certain very special cases involving constant distributions are solved. A necessary condition for a steady state is deduced in terms of an integral equation, in general non-linear.     

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.     

An analysis of the 2o and 10o field color-matching functions

The 2^o and 10^o field color-matching functions are independent: one specification is not a linear transformation of the other, even after correcting for macular pigment effects. Therefore, the “true” color-matching functions which directly describe the linear responses of the eye must be different for the two field sizes. This means that a given stimulus will, in general, have a different chromaticity depending upon the field size, regardless of the choice of any one colorimetric co-ordinate system for all field sizes. However, in spite of these chromaticity differences, a large uniform field usually appears nearly uniform. Such color uniformity implies that even though chromatic differences occur as a function of retinal position or field size, these differences are small. If this is the case, then the underling “true” color-matching functions determining the observed color-matching functions must be nearly, but not quite, identical. These differences vanish as identity between the sets of color-matching functions is approached. This property suggests a method of calculating the “true” color-matching functions. The “true” color-matching functions must approximate those obtained by minimizing the chromaticity differences between two independent sets of data. This can be done by assuming that the coefficients of transformation should be adjusted so as to produce as nearly identical chromaticities for spectrum stimuli as possible. In this paper, it is also assumed that the “true” color-matching functions have no negative values, as if they were based on actual absorption spectra. This article describes the calculation of the “true” 2^o and 10^o field color-matching functions satisfying these two conditions. For both field sizes, the maxima of the three functions are near 435, 540, and 585 mμ, after correcting for the filtering effects of the ocular media and macular pigment.     

A contribution to the mathematical biology of automobile driving: II. Passing as a case of psychophysical discrimination

The decision to pass or not to pass in view of an oncoming car is considered as a case of comparative judgment in which it is to be decided whether the time it will take to pass safely is greater or less than the time it will take to collide with the oncoming car. H. D. Landahl's well-known theory of psychophysical discrimination is used, and it is assumed that the “distracting stimuli” considered previously (Rashevsky, 1959,Bull. Math. Biophysics,21, 375–85) tend to increase the standard deviation of Landahl's fluctuation function. Effects of the “distracting stimuli” on the threshold of the neuroelements in Landahl's circuit are also considered. On this basis an expression is derived which gives the probability of a collision accident in passing as a function of the “distracting stimuli.”     

The Role of Organic Acids in the Domestication of <Emphasis Type="Italic">Oxalis tuberosa</Emphasis>: A New Model for Studying Domestication Resulting in Opposing Crop Phenotypes<Superscript>1</Superscript>

The Role of Organic Acids in the Domestication of Oxalis tuberosa: A New Model for Studying Domestication Resulting in Opposing Crop Phenotypes. Though few crops display directly opposing domesticated phenotypes, these crops may be the key to understanding domestication processes that address conflicting selective pressures in the agricultural ecosystem. Two relatively well-known examples are cassava (Manihot esculenta Crantz), which has high-cyanide and low-cyanide varieties, and potato (Solanum section Petota). Among the potatoes are several species, including the common potato (Solanum tuberosum L.), that have low levels of glycoalkaloids and there are other species of “bitter potato” with elevated levels of glycoalkaloids. We propose that Oxalis tuberosa Molina, “oca,” may represent a third example of such a crop system, with opposing high organic acid and low organic acid cultivars. Each cultivar set has different cultural food preparation practices (“use-categories”), similar to the “use-categories” that have been described for potatoes in the Andes (Brush et al. Economic Botany 35;70–88, 1981; Zimmerer Journal of Biogeography 18;165–178, 1991). Our initial analyses suggest that organic acids in tubers may be an important biochemical difference between use-categories, based on both oxalic acid and pH data. Here, we review our understanding of organic acids in oca tubers, while highlighting areas that merit further investigation.     

Zornia subsessilis (Leguminosae: Papilionoideae: Dalbergieae), a new species from Serra do Cabral, Minas Gerais, Brazil

Zornia subsessilis, a new species from Serra do Cabral, Minas Gerais State, Brazil, is described and illustrated on the basis of field and herbarium studies. It belongs to Zornia section Zornia of subgenus Zornia and it appears to be restricted to “campo rupestre” (rocky fields) within the “cerrado” biome. This new species is characterized by its sessile to subsessile leaves, fruits with articles 4 to 5 mm long and with bristles up to 3.5 mm long.     

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.     

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.     

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.     

Traditional Knowledge and Management of <Emphasis Type="Italic">Feijoa (Acca sellowiana)</Emphasis> in Southern Brazil

Traditional Knowledge and Management of Feijoa (Acca sellowiana) in Southern Brazil. This paper investigates traditional knowledge of the use and management of Acca sellowiana in southern Brazil. Fifty-six informants from three rural communities were assigned to one of four subgroups (“maintainers,” “managers,” “cultivators,” or “users”) based on their responses regarding management and use of A. sellowiana. Traditional knowledge related to use of this species is widespread among rural residents, but traditional knowledge related to management is fragmented depending on whether one uses, manages, or cultivates the species. Knowledge held in rural communities suggests that A. sellowiana could play an expanded role in local economies as well as biodiversity conservation. We suggest that participatory research could stimulate greater local use as well as on-farm conservation of A. sellowiana.     

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.     

Contribution to the mathematical theory of contagion and spread of information: I. Spread through a thoroughly mixed population

An equation is derived from the spread of a “state” by contact through a thoroughly mixed population, in which the probability of transmission depends both on the over-all duration of the process and on the time an individual has been in the “state.” Cases in which this probability is a function of only one or the other of the two “times” are worked out. It is shown that in the case of dependence on “private time” alone the asymptotic value of the fraction of the population effected is the same as that derived by the random net approach.     

On the spread of information with time and distance

The transmission of some information or behavior pattern is treated as a flow of “particles” which execute random motions over a population of individuals and which may multiply or disappear. Equations are derived for the number density of these “particles” and from this is calculated the number of individuals through which the “particles” have passed. The results are applied to a number of situations such as 1) uniform spatial distribution with multiplication factor decreasing with time because of loss of interest or confusion of the information, 2) multiplication factor constant but the rate of spreal decreasing with multiple hearings, 3) one-dimensional region with a small starting region with or without an absorbing barrier 4) two-dimensional region with absorbing barrier, 5) continous sources of information within a small region in one dimension, 6) uniform spatial distribution in which individuals do not respond to more than one hearing.     

Analysis of the exponential decay model of the neuron showing frequency threshold effects

The exponential decay model of a neuron has been analyzed using the “random walk” approach of stochastic processes and an “absorbing barrier” solution is obtained forg ^T (s)—the Laplace transform of the output pulse interval density function. An expression for the mean output frequency is derived from this and a variety of input-output curves plotted which show frequency threshold effects in single neurons. Our results are compared with those of other authors obtained by computer simulation techniques, and the significance of these results discussed with reference to the possible behavior of networks constructed of such neuron units.     

Word organization in coding DNA: A mathematical model

This article deals with the relationship between vocabulary (total number of distinct oligomers or “words”) and text-length (total number of oligomers or “words”) for a coding DNA sequence (CDS). For natural human languages, Heaps established a mathematical formula known as Heaps' law, which relates vocabulary to text-length. Our analysis shows that Heaps' law fails to model this relationship for CDSs. Here we develop a mathematical model to establish the relationship between the number of type of words (vocabulary) and the number of words sampled (text-length) for CDSs, when non-overlapping nucleotide strings with the same length are treated as words. We use tangent-hyperbolic function, which captures the saturation property of vocabulary. Based on the parameters of the model, we formulate a mathematical equation, known as “equation of word organization”, whose parameters essentially indicate that nucleotide organization of coding sequences are different from one another. We also compare the word organization of CDSs with the random word distribution and conclude that a CDS is neither similar to a natural human language nor to a random one. Moreover, these sequences have their unique nucleotide organization and it is completely structured for specific biological functioning. IM and AS contributed equally to this work.     

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.     

The Poetics of Purpose

Hackles have been raised in biosemiotic circles by T. L. Short’s assertion that semiosis, as defined by Peirce, entails “acting for purposes” and therefore is not found below the level of the organism (2007a:174–177). This paper examines Short’s teleology and theory of purposeful behavior and offers a remedy to the disagreement. Remediation becomes possible when the issue is reframed in the terms of the complexity sciences, which allows intentionality to be understood as the interplay between local and global aspects of a system within a system. What is called “acting for purposes” is not itself a type of behavior so much as a relationship between a dynamic system that “exists for a purpose” and its microprocesses that “serve purposes.” The “intentional object” of philosophy is recast here as the holistic self-organized dynamics of a system, which exists for the purpose of self-maintenance, and that constrains the parts’ behaviors, which serve the purpose of forming the system. (A “system” can be any emergent, e.g. an abiotic form, an adapted species, a self, a conditioned response, thought, or a set of ideas.) The self-organized whole, which is represented to the parts in their own constrained behaviors, assumes the guiding function so long attributed to the mysterious “intentional object.” If emergent self-causation is not disallowed, creative originality, as well as directionality, becomes part of the definition of purposeful behavior. Thus, key tools used here, required for understanding emergence, come from poetics rather than semoitics. In the microprocesses of self-organization, I find what I call “accidental” indices and icons — which are poetic in the sense that they involve mere metonymic contiguity and metaphoric similarity — and which are preferentially selected under constrained conditions allowing radically new connections to habituate into an “intentional” self-organized system that, not coincidentally, has some of the emergent characteristics of a conventional symbolic system.