Characterizing conditions for generalized Verhulst logistic growth of a biological population

It is shown that a biological population or subpopulation composed ofN individuals can be governed by a generalized Verhulst logistic equation with time-dependent rate functions if and only if certain characterizing conditions are satisfied by ∂N/∂N ₀ whereN ₀ is the value ofN att=0.     

Studies in imitative behavior: A generalization of the rashevsky model; Its mathematical properties

This paper is based on N. Rashevsky's theory of imitative behavior, the underlying idea being that performance of one reaction by a given individual produces an increased stimulation (or tendency) toward the same reaction in other individuals. For simplicity, consideration is limited to cases in which each individual may choose only between two (or two main categories of) reactions, denoted byA andB in the following. However, upon suggestion from Dr. Rashevsky and certainly in better agreement with actual facts, the strength of imitative interaction is assumed to vary from individual to individual. More precisely, if Ψ_i denotes the additional excitation caused by imitation in theith individual,PA_i the probability for performance of reactionA, andPB_i the probability for performance of reactionB by theith individual, we postulate that where the constants α _ik ^′ and β _ik ^′ (coefficients of imitative interaction) measure the amount of imitative influence exerted by thekth individual upon theith,N being the total number of individuals in the population. The term — α_i Ψ_i accounts for the spontaneous decay of excitation, and the quantities α _ik ^′ and α _ik ^′ are assumed to benon-negative. The expressions forPA_i andPB_i are obtained from H. D. Landahl's theory of conflicting stimuli; they depend non-linearly on the values Ψ_i. It is implicit in this formulation that the theory can only be applied if the frequency of contacts between individuals is not too small. Some further shortcomings and limitations of the model are outlined, and the discussion includes suggestions for reinterpretation and improvement of the theory. If all the quantities α _ik ^′ and α _ik ^′ have the same value, sayA, we return to the case treated by Rashevsky (and Landau, 1950); these authors, however, replace the sums in the equation above by integrals, which automatically restricts the validity of their results to very large values ofN. Their work may therefore be characterized by the assumption of uniform interaction in large populations. Our equations, on the other hand, are applicable even to very small groups, and therein lies one of their main advantages. In this paper the mathematical properties of the non-linear system of equations above are studied with particular reference to the existence and stability of steady states [dΨ_i/dt ≡ 0;, i = 1 , 2, . . . N]. A sufficient condition for the existence of only one stable steady state is derived. It may be formulated roughly by stating that all the coefficients of interaction should be sufficiently small. It that is not the case, there may exist a greater number of stationary states. In particular, two of them (called “extremal”) have the following properties: they arestable and such that the average number of individuals in the group performing one or the other reaction is the largest (or smallest) possible as compared with the other steady states. Hence the situation is qualitatively similar to that found by Rashevsky and Landau.Quantitatively, however, important differences may arise, depending on the nature of the matrix specifying the interaction. A stable state may be approached through damped oscillations, but this effect is important only if the damping is sufficiently small for the oscillations to become practically observable. Little information could be obtained on this point, due to mathematical difficulties. As mentioned above, the most interesting applications of this theory will be with respect to small populations or to populations partitioned into subgroups with varying amounts of imitative interaction within as well as between groups.     

Mathematical biosociology of beliefs and prejudices

Following a suggestion made previously (Bull. Math. Biophysics,13, 61, 1951), it is assumed that every individual has both a tendency to behavearationally, by accepting everything on faith, and rationally, by subjecting everything to rational analysis. Arational behavior is characterized by various beliefs, prejudices, etc., which are considered to be conditioned reactions, learned by the individual before he completely develops his faculties for rational thinking. The two tendencies are assumed to be due to excitations of two different regions of the central nervous system, and are measured by the intensities ɛ _f and ɛ _r of those excitations. Those intensities are further assumed to increase linearly with time, the increases of the two beginning, in general, at different ages. The rates of increase are considered as normally distributed in the population. The relative frequency of arational and rational behavior is determined by the difference φ=ɛ _f =ɛ _r according to equations 0 developed previously (Bull. Math. Biophysics,11, 255, 1949). It is shown that with the above assumptions the majority of the population, which starts with arational behavior, will, within two or three of generations, either change to rational behavior or continue indefinitely to behave arationally. This will hold as long as imitative factors are present. Expressions for the numbers of individuals who behave rationally and arationally are derived. If the intensity of conditioning toward an arational behavior decreases with increasing size of the rationally behaving minority, or, if the rationally behaving individuals are not influenced by imitation, then a slow secular trend toward rational behavior may be present. An expression is also derived for the fraction of individuals who behave rationally as a function of age. This fraction increases with increase of the age at which the beginning conditioning toward any beliefs or prejudices begins.     

Genetic variation, genetic structure and effective population size in the tropical holoparasitic endophyteBdallophyton bambusarum (Rafflesiaceae)

The genetic population structure inBdallophyton bambusarum, an endoparasite, was studied in ten subpopulations from a subdeciduous tropical forest in Veracruz Mexico. The sample was analyzed using seven polymorphic loci in cellulose acetate electrophoresis. Isozyme data indicated that the subpopulations ofB. bambusarum contained high genetic variability (H_ep = 0.452 ± 0.045, S.E.). Our analysis suggests that almost each inflorescence ofB. bambusarum is an individual. The subpopulations studied were genetically similar (average Nei's genetic identity 0.941 ± 0.051 and F _st values 0.097 ± 0.026), suggesting that genetic differentiation among subpopulations was small. Direct estimates of effective population size was derived from observations of three fluorescent dyes, and from the genetic neighborhood area derived from these data. The neighborhood area, multiplied by the total density of individuals, gave an N_e = 124.84 plants, and when corrected to consider the proportion of males and females gave an N_e = 118.59 individuals. An indirect estimate of Nm was obtained from the F _st values (mean Nm=2.037), giving an indirect estimate of the effective population size N_b = 12.8 individuals. Both values are relatively high when compared to other plant studies. The gene flow and/or effective populations size of the studied subpopulations ofB. bambusarum are believed to be large enough to prevent differentiation among subpopulations due to genetic drift.     

Artifacts associated with the use of thiocyanate and valinomycin/K+ as permeant ions in oxidant pulse experiments on denitrifying bacteria

By means of¹⁵N-tracer and oxidant pulse methods and with nitrate-grownParacoccus denitrificans it was found that KSCN completely inhibited reduction of N₂O and nitrate in the 1 to 10 mM range, but had little or no effect on reduction of O₂ or nitrite at 150 mM. These observations confirm a previous report. Potassium thiocyanate was insufficiently permeant across the cytoplasmic membrane ofParacoccus denitrificans andPseudomonas denitrificans even at 150 mM to prevent membrane polarization when oxidant pulses were large. Polarization and inhibition artifacts due to KSCN have created some confusion in the literature. Whereas valinomycin had little direct effect on reduction of nitrite, N₂O, and O₂ individually byPa. denitrificans, it caused a temporary nitrite-dependent inhibition of N₂O reduction. Under nonpolarizing conditions the H⁺/2e^– ratios for O₂, N₂O, and nitrite (8.0, 4.3, and 4.5, respectively) confirmed those reported previously from this laboratory. The present results largely but not entirely agree with data from another laboratory.     

Ferrioxamine transport mutants and the identification of the ferrioxamine receptor protein (FoxA) inErwinia herbicola (Enterobacter agglomerans)

Summary Iron deprivation ofErwinia herbicola (Enterobacter agglomerans) induces the biosynthesis of six high-M _r outer-membrane proteins and large amounts of ferrioxamine E. Mutagenesis withN-methyl-N-nitro-N-nitrosoguanidine and selection with ferrimycin A yielded mutants ofE. herbicola K4 (wild type), defective in the expression of a 76-kDa outer-membrane protein, as determined by SDS/polyacrylamide gel electrophoresis. While in bioassays wild-type cells showed growth promotion in the presence of ferrioxamines (B, D₁, D₂, E, G), enterobactin, citrate, ferrichrome and coprogen, these mutants failed to respond to ferrioxamines. Moreover, experiments with⁵⁵Fe-labelled siderophores confirmed that iron transport mediated by ferrioxamine E and B in the mutants was completely inhibited, whereas iron transport by other hydroxamate siderophores, such as ferrichrome and coprogen was unaffected. The results are evidence that the 76-kDa protein in the outer membrane represents the receptor protein (FoxA) for ferrioxamines inE. herbicola.     

Denitrification of nitrate to nitrogen gas by washed cells ofRhizobium japonicum and by bacteroids fromGlycine max

Nitrate, nitrite and nitrous oxide were denitrified to N₂ gas by washed cells ofRhizobium japonicum CC706 as well as by bacteroids prepared from root nodules ofGlycine max (L.) Merr. (CV. Clark 63). Radiolabelled N₂ was produced from either K¹⁵NO₃ or Na¹⁵NO₂ by washed cells ofRh. japonicum CC705 grown with either nitrate only (5 mM) or nitrate (5 mM) plus glutamate (10 mM). Nitrogen gas was also produced from N₂O. Similar results were obtained with bacteroids ofG. max. The stoichiometry for the utilization of¹⁵NO ₃ ^- or¹⁵NO ₂ ^- and the produciton of¹⁵N₂ was 2:1 and for N₂O utilization and N₂ production it was 1:1. Some of the¹⁵N₂ gas produced by denitrification of¹⁵NO ₃ ^- in bacteroids was recycled via nitrogenase into cell nitrogen.     

Electrical responses to two clicks: A simple statistical interpretation

A probability model for the amplitude of the first neural component,N ₁, of the click response has been developed on the assumption thatN ₁ is the summated effect of practically simultaneous firing from large numbers of independent neural units. Equations derived from the model predict the course of recovery for the response to the second of two clicks. A mathematical relation is shown between the two-click situation and the intensity growth curve for the first neural component of the response to a single click. This permits indirect reconstruction of the function relating amplitude ofN ₁ to stimulus intensity. Experiments with pairs of clicks indicate that a normal probability integral provides a satisfactory fit. Thus the entire intensity function ofN ₁ is capable of specification in terms of two statistical parameters, the mean and the standard deviation of the normal integral. Factors that affect the click response are presumed to act upon these parameters. This research was carried out under Contract N5ori-76 between Harvard University and the Office of Naval Research, U. S. Navy (Project Nr142-201, Report PNR-107). Reproduction for any purpose by the U. S. Government is permitted.     

Light-illumination effects on the cellular concentration of photolyase molecules in yeast

Summary The effects of light-illumination in the absence or presence of cycloheximide (CHX) on the number of photoreactivating enzyme molecules per haploid cell ofSaccharomyces cerevisiae (N _PRE) were investigated.N _PRE increased, when the cells were held in buffer in light in the absence of CHX for 24 or 48 h before UV-irradiation (buffer-holding). The increase ofN _PRE was clearly observed in cells, whoseN _PRE before bufferholding was small, that is, in stationary growth phase cells grown at 37° C and in logarithmic growing cells incubated at 30° C. In the case of buffer-holding in light in the presence of CHX or in the dark in the absence or presence of CHX, the increase ofN _PRE was small or not observed.     

Survival of ciliate protozoa under starvation conditions and at low bacterial levels

Under starvation conditions, 50% survivorship times displayed no significant relationship with cell size in 2 ciliate species in this study and 5 protozoan species from the literature. Differences in survival ability were attributed to differences in weight-specific respiratory rate and relative motility among these 7 species. At low bacterial levels, 4 ciliate species in this study displayed significant differences in survivorship. High survivorship ofEuplotes patella relative to that ofParamecium caudatum andParaurostyla sp. at low ciliate densities was attributed to the lower individual energy requirements of this smaller species. High survivorship ofStentor coeruleus was interpreted as an effect of its large quantity of reserves and low respiratory rate. The survivorship ofE. patella was reduced at a higher population density. Four ciliate species survived longer at 15C than at 22C. Q₁₀ values based on 50% survivorship times at these 2 temperatures were much lower than Q₁₀ values based on respiratory rates and growth rates of well-fed ciliates over a similar temperature range.     

Effect of vesicular-arbuscular mycorrhizal fungi and phosphate-solubilizing bacteria on growth and nutrition of soybean in a neutral-calcareous soil amended with32P-45Ca-tricalcium phosphate

Summary The optimum conditions for growth ofFrankia sp. HFPCcI3 isolated fromCasuarina cunninghamiana, were studied in batch culture using defined media. Maximum growth (doubling time was 24 h)_was achieved at 33°C and at pH 6.3 with pyruvate and NH ₄ ⁺ as the sole C and N sources, respectively. Removal of NH ₄ ⁺ from the culture medium resulted in vesicle differentiation which was paralleled by induction of acetylene reduction activity. Growth on atmospheric N₂ was optimal with combined pyruvate and glucose as the carbon sources and displayed a doubling time of about 48 h. Comparisons in growth and N₂-fixing activity ofFrankia strains grown in a variety of cultural conditions demonstrate the range of behavior among the strains.     

Control of sodium permeability of the outer barrier in toad skin

Summary The²⁴Na efflux (J _eff ^Na ) (i.e., the rate of appearance of²⁴Na in the outer compartment) in the isolated short-circuited toad skin bathed by NaCl-Ringer's solution on both sides is composed of para- and transcellular components of almost equal magnitudes. This relies on the assumption that amiloride acts on the transcellular component only and could block it completely.Ouabain induces a large transient increase of the transcellular component. This increase, which starts within a few minutes after the addition of ouabain, is due to electrical depolarization of the outer barrier, rather than a consequence of blocking Na recirculation across the inner barrier. The subsequent decline ofJ _eff ^Na , which takes place after the ouabain-inducedJ _eff ^Na peak, is due to a progressive block of outer barrier Na channels with time, which can eventually be complete, depending on the duration of action of ouabain. As the external Na concentration was always kept high and constant in these experiments, the results indicate that a rise in cell Na concentration, and not in the outer bathing solution, is the signal that triggers the reduction of outer barrier Na permeability (P ₀ ^Na ).Ouabain has no effect uponJ _eff ^Na with Na-free solution bathing the outer and NaCl-Ringer's solution the inner skin surface, showing the importance of Na penetration across the outer barrier, and not across the inner barrier due to its low Na permeability, in the process of closing the Na channels of this structure.Step changes from Na 115mm to Na-free external solution, or vice-versa, may affect both the outer barrier electrical potential difference (PD₀) and cell Na concentration (Na) _c . Therefore, the behavior ofJ _eff ^Na depends on which variable (if PD₀ or (Na) _c regulated outer barrier Na permeability) is most affected by step changes in outer bathing solution Na concentration.Amiloride in the control condition blocks the transcellular component ofJ _eff ^Na . However, in the condition of approximate short-circuiting of the outer barrier and high cellular Na concentration induced by long term effects of ouabain, when the Na channels of the outer barrier are already blocked by elevated cell Na concentration, amiloride may induce the opposite effect, increasing Na permeability of the outer barrier.With outer barrier Na channels completely blocked by high cell Na concentration, PCMB in the outer bathing medium induces a large increase ofJ _eff ^Na , rendering these channels again amiloride sensitive.The results are consistent with the notion that Na efflux from cell compartment to the outer bathing solution goes through the amiloride-sensitive Na channels of the apical border of the superficial cell layer of toad skin, with an apparent Na permeability modulated by cell ionic environment, most probably the cell Na concentration.The ensemble of the present results are consistent with Na permeability regulation taking place at the outer barrier level. However, this precise location could only be made unambiguously by measurements across the individual outer cell membranes.     

II. Photoaffinity probes provide a general method to prepare peptide-conjugates from native protein fragments

The advantages of using amino-directed photoprobes to couple native fragments, obtained by enzymatic digestion with trypsin, to protein carriers to prepare peptide-conjugates is described. The following photoprobe reagents were investigated:N-hydroxysuccinimidylp-azidobenzoate,N-hydroxysuccinimidyl ester ofp-azidobenzoylglycine,N-hydroxysuccinimidylp-benzoylbenzoate, and pentachlorophenyl ester ofp-benzoylbenzoyl glycine or the symmetric anhydride ofp-benzoylbenzoylglycine. These reagents modify only the NH₂-terminal amino group and/or COOH-terminal -amino group of lysine of the tryptic fragments. Since the photoprobe is inert until photolysis, the probe-modified native fragment can be readily purified by high-performance liquid chromatography before cross-linking to the carrier molecule. The benzophenone photoprobes were shown to give the highest incorporation of peptide onto the protein carrier.     

Cadmium-copper antagonism in the activation of periplasmic nitrous oxide reductase of copper-deficient cells fromPseudomonas stutzeri

Summary Copper-deficient cells ofPseudomonas stutzeri strain ZoBell synthesize catalytically inactive nitrous oxide (N₂O) reductase which is activated by added Cu(II) in the absence of de novo protein synthesis. The apparentK _m for the activation process is 0.13 M. Activation is temperature-dependent and is inhibited by Cd(II)(K _i 1.27 M) and less strongly by Zn(II), Ni(II), and Co(II). The same metal ions at 20 M have little or no effect on N₂O reduction of intact cells. Apo-N₂O reductase of transposon Tn5-inducednos ^– mutants with defective Cu-chromophore biosynthesis is not reactivated by Cu(II). N₂O reductase of Cu-sufficient and Cu-deficient wild type, and ofnos ^– mutants is localized in the periplasm, the latter providing the likely site of metal incorporation into the apoenzyme.     

Skewed reproductive success among male white-spotted charr land-locked by an erosion control dam: Implications for effective population size

In 1996 and 1997, the spawning behavior of fluvial white-spotted charr Salvelinus leucomaenis, was observed in the upstream area of an erosion control dam. A small number of males with relatively large body size mated successfully with females as a pair, while almost all satellite males did not sneak successfully, resulting in a non-random mating system. The low sneaking success of subordinate males, in addition to the monopolization of spawning opportunities by a few dominant males, is one of the most important causes of skewed reproductive success among males. The total number of adult fishes in the study area (N: approximately half of the whole tributary above a dam) was estimated as 148 and 102 in 1996 and 1997, respectively. Based on these findings and some further assumptions, the estimated effective population size (N_e) was low in both years. The N_e/N ratio ranged from 0.33 to 0.36 in both years. In addition to reduced population size by construction of an impassable dam, the above-dam population suffered low N_e due to skewed reproductive success among males. The low N_e may be one cause of extinction in above-dam populations of fluvial charr, especially just after the construction of impassable barriers.     

Simple method for calculating confidence limits around inbreeding rate and effective population size estimates from pedigrees

An important concept in population genetics is effective population size (N_e), which describes the expected rate of loss of genetic variability from a population. One way to estimate N_e is using a pedigree. However, there are no methods for comparing the N_e estimated from a pedigree with that expected from life-history models. In the paper we show how N_e can be estimated from the change in inbreeding rate (f) estimated from a pedigree. The mean individual inbreeding rate in a population at a given time must be calculated from averaged values for males and females, where each age class is weighted by its reproductive value. We show an exact method for placing confidence intervals around f and N_e using a binomial distribution, and present a method for approximating this interval for large N_es using a Poisson distribution. These confidence intervals can be used to compare f and N_e from a pedigree to expected values from demographic models, and to compare N_es of two populations.     

Genetic specificity of nitrogen nutrition in leguminous plants

Summary Mineral nitrogen did not increase grain yield and seed protein levels ofVicia faba L. andLupinus luteus L. in field trials and pot experiments. Fixed N₂ was substituted by mineral nitrogen in these cases because of inhibition of N₂ fixation by mineral nitrogen. Contrary to these results mineral nitrogen increased grain yields and seed protein amounts ofLupinus albus L.,Pisum sativum L., andGlycine max. (L.) Merr. The nitrogen effect was caused at an early stage by saving energy due to inhibition of N₂ fixation (measurement of gas exchange by means of IRGA). In case of the N application after flowering grain, yields and seed protein levels increased because the mineral N was an additional nitrogen source for plants. At this stage the plants had ceased fixing atmospheric nitrogen. The high sink activity of growing fruits induced a lack of assimilates in nodules (determined by means of¹⁴CO₂ application). The N effect was therefore the consequence of the lower assimilate pool for supplying root nodules in these plants in comparison withVicia faba L. andLupinus luteus L. Hence it follows that response to mineral nitrogen can be a criterion for discovering more effective Rhizobium-host combinations.     

Dinitrogen fixation (C2H2 reduction) by bacterial strains at various temperatures

Acetylene-reducing activities (ARA) of strains ofEnterobacter agglomerans, Azospirillum brasilense, Azotobacter chroococcum, and Bacillus, isolated from temperate or tropical soils, were compared at different temperatures to study temperature adaptability. All Enterobacter strains and Bacillus strain C-11-25 reduced C₂H₂ at temperatures as low as 5°C. ARA by Enterobacter strains declined sharply above 30°C but ARA by Bacillus strain C-11-25 continued to increase with an increase in temperature.A. brasilense strain sp 245, isolated from wheat roots in Brazil, reduced more C₂H₂ at lower temperatures than strain Cd, isolated from a Californian soil. Similarly, the temperate strain ofA. chroococcum was a better N₂ fixer than the tropicalA. chroococcum strain at lower temperatures. Tropical strains ofA. brasilense andA. chroococcum reduced more C₂H₂ than temperate strains at higher temperatures. Therefore, it appears that temperate and tropical N₂-fixing organisms adapt themselves to their particular environment and should have more potential to benefit crops grown at the particular temperatures favorable to them. Only Bacillus strain C-11-25 has potential to benefit both temperate and tropical crops because it reduced significant acetylene over a wide temperature range.