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.     

Mathematical biophysics of automobile driving

In a previous paper (Rashevsky, 1959,Bull. Math. Biophysics,21, 299–308) we derived an approximate expression for the maximal speed of driving in terms of the reaction time of the driver. In the present paper the possible effects of unevennesses of the pavement, of such distracting stimuli as road signs etc. on the reaction time are studied theoretically, using previous developments of the mathematical biophysics of the central nervous system. In this manner expressions are derived which determine the maximal safe speed in terms of road conditions and other distracting stimuli. Effects of those conditions on fatigue are also discussed.     

Mathematical biology of automobile driving: I. The shape of the tracking curve on an empty straight road

The idea was suggested by the author previously (Bull. Math. Biophysics,22, 257–262, 1962) that the keeping of the car close to the center of the lane is a problem of psychophysical discrimination between two conflicting stimuli, namely a stimulus to turn away from the left, resp. right edge of the lane. This is elaborated in the present paper. The effects of discrimination threshold and of the endogenous fluctuations which result in erroneous judgments are discussed. In order that driving should be possible at all, a relation, derived in this paper, must hold between the threshold of discriminationh, the sensitivity coefficientb of the driver to changes in the distance between the car and the edge of the lane, and the width of the lane. General expressions are derived which characterize the stochastic nature of the tracking curve.     

Bacterial communication through membrane vesicles

ABSTRACT

Bacteria can communicate through diffusible signaling molecules that are perceived by cognate receptors. It is now well established that bacterial communication regulates hundreds of genes. Hydrophobic molecules which do not diffuse in aqueous environments alone have been identified in bacterial communication, that raised the question on how these molecules are transported between cells and trigger gene expressions. Recent studies show that these hydrophobic signaling molecules, including a long-chain N-acyl homoserine lactone signal produced in Paracoccus denitrificans, are carried by membrane vesicles (MVs). MVs were thought to be formed only through the blebbing of the cell membrane, but new findings in Pseudomonas aeruginosa and Bacillus subtilis revealed that different types of MVs can be formed through explosive cell lysis or bubbling cell death, which findings have certain implications on our view of bacterial interactions.     

INBREEDING AND VARIANCE EFFECTIVE SIZES FOR NONRANDOM MATING POPULATIONS

Following an inbreeding approach and assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, I have derived expressions for the inbreeding effective size, N_eI, for a finite diploid population with variable census sizes for three cases: monoecious populations with partial selfing; dioecious populations of equal numbers of males and females with partial sib mating; and unequal numbers of males and females with random mating. For the first two cases, recurrence equations for the inbreeding coefficient are also obtained, which allow inbreeding coefficients to be predicted exactly in both early and late generations. Following the variance of change in gene frequency approach, a general expression for variance effective size, N_eV, is obtained for a population with unequal numbers of male and female individuals, arbitrary family size distribution, and nonrandom mating. All the parameters involved are allowed to change over generations. For some special cases, the equation reduces to the simple expressions approximately as derived by previous authors. Comparisons are made between equations derived by the present study and those obtained by previous authors. Some of the published equations for N_eI and N_eV are shown to be incomplete or incorrect. Stochastic simulations are run to check the results where disagreements with others are involved.     

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.     

Further contributions to the mathematical biophysics of automobile driving

The discussions of a previous paper (Bull. Math. Biophysics,21, 299–308, 1959) are generalized by considering that the angular direction error made by the driver, as well as the driver's reaction time are not constant but are randomly distributed. Instead of a critical speed, at which the car will jump off the road, we now find that for every speed there is a probability of the car to jump off the road but that this probability is vanishingly small for sufficiently low speeds, yet increases rapidly for high speeds. Thus a more realistic picture of the process of driving is obtained. When the standard deviation of the distribution functions for the angle and the reaction time are very small, the expression obtained here reduces to the expression obtained previously.     

Burden of cancer attributable to tobacco smoking in member countries of the Association of Southeast Asian Nations (ASEAN), 2012

BackgroundCancer is an increasing problem in ASEAN (Association of Southeast Asian Nations). Tobacco use is a well-established risk factor for many types of cancers. Evidence on burden of cancer attributable to tobacco is essential to raise public and political awareness of the negative effects of tobacco on cancer and to be used to stimulate political action aims at reducing smoking prevalence in ASEAN member countries. The objective of this study was to estimate burden of cancer attributable to tobacco smoking in ASEAN, 2012.MethodsIn this study, smoking prevalence was combined with Relative Risks (RRs) of cancer to obtain Smoking Attributable Fractions (SAFs). Cancer incidence and mortality data among individuals aged 15 years and older were derived from GLOBOCAN 2012. Fourteen types of cancer were included in the analysis. Sensitivity analyses were conducted to examine the impact of the use of alternative RRs and the use of alternative prevalence of smoking in some countries.ResultsThe findings showed that tobacco smoking was responsible for 131,502 cancer incidence and 105,830 cancer mortality in ASEAN countries in 2012. In other words, tobacco smoking was accounted for 28.4% (43.3% in male and 8.5% in female) of cancer incidence and 30.5% (44.2% in male and 9.4% in female) of cancer mortality in ASEAN. When looking at the types of cancer, lung cancer showed the strongest association with tobacco smoking. Incidence of cancer and cancer mortality attributable to tobacco smoking varied by countries due to the differences in size of population, background risk of cancer, and prevalence of smoking in each country. According to the sensitivity analyses, RRs of lung cancer, pharynx cancer, and larynx cancer used in the estimates have significant impact on the estimates.ConclusionsAs about one-third of cancer incidence and mortality in ASEAN are attributable to tobacco smoking ASEAN member countries are strongly encouraged to put in place stronger tobacco control policies and to strengthen the existing tobacco control measure in order to effectively control cancer.     

Calculation of the cumulative distribution function of the time to a small observable tumor

Multistage mathematical models of carcinogenesis (when applied to tumor incidence data) have historically assumed that the growth kinetics of cells in the malignant state are disregarded and the formation of a single malignant cell is equated with the emergence of a detectable tumor. The justification of this simplification is, from a mathematical point of view, to make the estimation of tumor incidence rates tractable. However, analytical forms are not mandatory in the estimation of tumor incidence rates. Portier et al. (1996b, Math. Biosci. 135, 129–146) have demonstrated the utility of the Kolmogorov backward equations in numerically calculating tumor incidence. By extending their results, the cumulative distribution function of the time to a small observable tumor may be numerically obtained.     

On the equilibrium distribution of population in space

Spatial equilibrium distributions of population are derived from the spatial distribution of net rates of reproduction, and from a relationship between migratory flow and gradients of population density and of locational “attractiveness.” Conditions are discussed for which population approaches a uniform spatial density. Under certain conditions a particularly simple statement of the equilibrium conditions is possible in terms of the “potential of population,” a concept introduced by demographers (J. Q. Stewart,Geographical Review,37, 46–85, 1947) to measure the proximity of a point to people. This paper was first written at the Center for Advanced Study in the Behavioral Sciences.     

On a restricted occupancy model and its applications

With the multivariate hypergeometric distribution as a background certain occupancy distributions useful in practical applications are derived. More specifically it is assumed that a sample of n individuals is drawn from a population consisting of m types with r individuals in each type, (i) without replacement and (ii) by returning the selected individual in the population and with it another individual of the same type. The distributions of the number Z of distinct types observed in the sample are obtained in both cases in terms of the numbers. Assuming, in addition to the m equiprobable types of individuals, the existence of a control type, say, with s individuals, the joint distribution of the number U of distinct types observed in the sample and the number V of individuals of the control type present in the sample is obtained in terms of the numbers C(n, k, r) and the marginal distribution of U in terms of the Gould-Hopper numbers. Using these distributions minimum variance unbiased estimators of the number m of types are derived. Moreover small sample tests based on the zero frequency are constructed.     

Contributions to the mathematical biophysics of the central nervous system with special reference to learning

A learning theory based on the lowering of thresholds of neurons under certain conditions is applied to two “random net” models. The first, a so-called “ganglion-brain” is characterized by completely random connections of all afferent tracts except certain ones which form the pathways for unconditioned responses. Certain expressions are derived which measure the learning potentiality of the ganglion— in particular, with respect to the number of responses which can be learned (conditioning potential) and the amount of interference between the learned responses (redundance potential). The second model concerns the progressive refinement of a response. The efficiency of learning in this case is reflected in the eventual specificity of the response which, in turn, depends on the modification of the distribution of thresholds associated with the neurons governing the responses. Expressions are derived relating the initial distribution of thresholds, the relative effectiveness of the various responses, and certain other parameters to the final distribution of thresholds. For a particular choice of the effectiveness distribution of responses the progressive sharpening of the threshold curve (i.e., progressive specificity of response) is demonstrated. Some implications of the model with respect to the evolution of nervous systems are discussed.     

The basic flow equations of electrophysiology in the presence of chemical reactions: II. A practical application concerning the pH and voltage effects accompanying the diffusion of O2 through hemoglobin solution

Previous papers by F. M. Snell (Jour. Theor. Biol.,8, 469–479, 1965) and M. A. Fox and H. D. Landahl (Bull. Math. Biophysics,27, Spec. Issue, 183–190, 1965) have found that the formulation by previous authors for the oxygen flow rates through hemoglobin solution as a function of pressure determined by E. Hemmingsen and P. F. Scholander (Science,132, 1379–1381, 1960) did not give a satisfactory quantitative fit of the curve for constant pressure difference. The suggestion of Fox and Landahl that the Bohr effect involving the shift in acidity accompanying the oxidation of Hb should give rise to voltage and pH differences in oxyhemoglobin transport is examined in more detail. In this paper, the previous expressions for the total oxygen flow rate in terms of the end point concentrations are extended to include the effects of the electrical field. Estimates of the potential difference shows it to be negligible. A derivation of a voltage-pH relation shows that the Nernst relation does not apply and a negligible voltage difference does not preclude a pH shift which is the more probable explanation of the discrepancies observed. Several other predictions suitable for experimental testing are made.     

A field theory of neural nets: II. Properties of the field equations

The field equation derived in Part I (Griffith,Bull. Math. Biophysics,25, 111–120, 1963a) is examined further. The stability of critical solutions is investigated and it is shown that, at least in certain cases, general solutions tend toward critical solutions. The relationship between the present field theory and a conventional matrix formulation is derived.     

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.     

The motion of the left ventricle: II

We use the dimensional parameters previously derived (Bull. Math. Biophysics,28, 355–362, 1966), the ventricular pressure expressed as a Fourier series, and several additional assumptions to derive expressions for the mechanical parameters of the ventricle: flow, muscle segment length, surface area, transmural force, wall tension and work. The wall of the ventricle is assumed to consist of three layers of muscle. Each of the mechanical parameters is expressed in terms of Fourier series.     

Cytogenetics of New Zealand blackflies of the genus Austrosimulium (Diptera: Simuliidae) 1. The cytogenetics of Austrosimulium australense

Abstract

This study undertakes a cytogenetic analysis of the New Zealand blackfly species Austrosimulium australense (Schiner). The principles of such an approach are outlined, and previous taxonomic studies of the genus Austrosimulium, in particular the taxonomic position of A. australense, are discussed. Populations from North Island localities covering a wide area were sampled and analysed for polymorphisms in the polytene chromosomes, taken from salivary glands of larvae. In all, 1018 larvae from 49 sites were analysed. A polytene chromosome map of A. australense is presented, with details of chromosomal inversions found in salivary gland cells. Three geographical zones are designated, according to the incidence of certain chromosomal polymorphisms.     

“Pollination” of a fungus by a fly

Summary The mechanism resulting in fertilization of Epichloë typhina, a heterothallic ascomycete that is an endophytic pathogen of grasses, has now been discovered. Conidia of one mating type are produced in stromata and are then transferred by insects to individuals of the opposite mating type. One insect, Phorbia phrenione, is a particularly important vector of conidia. Once conidia of the opposite mating type have been transferred to a stroma, the life cycle continues with the formation of perithecia.     

A neurobiophysical interpretation of certain aspects of the problem of risks

A situation is considered in which an individual is given an opportunity to risk a certain amount of money or goods in order to gain a larger amount, providing the result of some uncertain event proves favorable. A neural mechanism is introduced in which the probability of success can be ordered according to its value. The response to a situation thus becomes dependent upon the probability. An expression is then derived for the amount that would be risked in terms of the probability and of the amount that would be gained in the event of a successful outcome. Similar expressions are obtained for the case of insurance against loss. Results of questionnaires indicate that individuals can be classified according to their pattern of behavior in the situation considered. The various types can be most easily recognized when a plot is made of the relative amount risked against the probability of success. In a general way, these types can be understood in terms of the equations derived.