The extinction of slowly evolving dynamical systems

The time evolution of slowly evolving discrete dynamical systems x _{i + 1} = T(r _i ,x _i ), defined on an interval [0, L], where a parameter r _ichanges slowly with respect to i is considered. For certain transformations T, once r _i reaches a critical value the system faces a non-zero probability of extinction because some x _j [0, L]. Recent ergodic theory results of Ruelle, Pianigiani, and Lasota and Yorke are used to derive a simple expression for the probability of survival of these systems. The extinction process is illustrated with two examples. One is the quadratic map, T(r, x) = rx(1 – x), and the second is a simple model for the growth of a cellular population. The survival statistics for chronic myelogenous leukemia patients are discussed in light of these extinction processes. Two other dynamical processes of biological importance, to which our results are applicable, are mentioned.     

Optimal Fixing of the Bandwidth-Parameter for the Empirical Regression1,2

The estimator ?₀(x) of the regression r(x) = E (Y | × = x) from measured points (x_i, y_i), i = 1(1) n, of a continuous two-dimensional random variable (X, Y) with unknown continuous density function f(x, y) and with moments up to the second order can be made with the help of a density estimation f?₀(x, y) (see e.g. SCHMERLING and PEIL, 1980). Here f?₀(x, y) still contains free parameters (so-called band-width-parameters), the values of which have to be optimally fixed in the concrete case. This fixing can be done by using a modification of the maximum-likelihood principle including jackknife techniques. The parameter values can be also found from the estimators for r(x). Here the cross-validation principle can be applied. Some numerical aspects of these possibilities for optimally fixing the bandwidth-parameter are discussed by means of examples. If ?₀(x) is used as a smoothing operator for time series the optimal choice of the parameter values is dependent on the purpose of application of the smoothed time series. The fixing will then be done by considering the so-called filter-characteristic of ?C₀(x).     

Generation of rules for expert systems by statistical methods of fermentation data analysis

Five new methods for determining the relations between kinetic data of fermentations are described and applied to an industrial antibotic fermentation process. The input data for these method are the elements of the distance matrix d_ij, which quantify the sum of the deviation squares between the time dependent kinetics x (t) of the fermentation runs i and j. For each measurable or calculable kinetic state variable, one n x n distance matrix must be calculated where n is the number of fermentation runs. All methods compare these distance matrices by statistical or graph-theoretical approaches. The algorithms obtained are universally applicable if enough kinetic data are available, especially from more than 10 comparable fermentation runs. The algorithms were developed for the use in knowledge acquisition modules of expert systems.     

Detecting and mapping repulsion-phase linkage in polyploids with polysomic inheritance

It has been suggested that ratios of coupling- to repulsion-phase linked markers can be used to distinguish between allopolyploids and autopolyploids, because repulsion-phase linkages are much more difficult to detect in autopolyploids with polysomic inheritance than allopolyploids with disomic inheritance. In this report, we analyze the segregation pattern of repulsion-phase linked markers in polyploids without complete preferential pairing. The observed repulsion-phase recombination fraction (R) in such polyploids is composed of a fraction due to crossing-over (R_c) and another fraction due to independent assortment (R_i). R_i is the minimum distance that can be detected between repulsion-phase linked markers. Because R_i is high in autopolyploids (0.3373, 0.4000, 0.4286 and 0.4444) for autopolyploids of 2n=4x, 6x, 8x and 10x), large population sizes are required to reliably detect repulsion linkages. In addition, the default linkage used in mapping-programs must be greater than the corresponding R_i to determine whether a polyploid is a true autopolyploid. Unfortunately, much lower default linkages than the R_is have been used in recent polyploid studies to determine polyploid type, and markers have been incorporated into polyploid maps based on the R values. Herein, we describe how mapping repulsion linkages can result in spurious results, and present methods to accurately detect the degree of preferential pairing in polyploids using repulsion linkage analysis. Received: 29 February 2000 / Accepted: 17 July 2000     

Fluctuations and Noise in Kinetic Systems: Application to K+ Channels in the Squid Axon

We consider the equilibrium or steady-state noise power density spectrum in the quantity N = Σ^x_i=0 a_iN_i for an ensemble of independent and equivalent systems each of which can exist in the discrete set of states i = 0, 1, ···, x. N_i is the number of systems of the ensemble in state i and the a_i's are constants. There is a transition rate constant α_ij for an arbitrary transition i → j; the kinetic equations are linear. There are possible applications to enzyme and biochemical kinetics generally, to membrane transport, muscle contraction, binding on macromolecules, etc. In each case, noise measurements would provide information about the kinetic scheme. The particular application considered here is to K⁺ channels or gates (one channel = one system) in the squid axon membrane: a_iK is the K⁺ conductance of a channel in state i and the kinetic scheme is of the Hodgkin-Huxley type (HH). Here we allow an arbitrary set of a_i's. This is a generalization of our treatment of K⁺ channel noise in an earlier paper. The theory is discussed and some calculations made using Fishman's recent experimental results on K⁺ channel noise as a guide. Preliminary indications are that the HH choice of a_i's may be oversimplified and that a₀ 0, a₁ ≠ a₀, a_x ≠ a_x-1. Quite possibly the a_i's increase from a₀ to a_x, though the early a_i's must be relatively small to give the observed induction behavior in g_K(t). An increase in equal steps is unsatisfactory because this is essentially HH with x = 1 (no induction). More refined experiments may modify these tentative conclusions. In any case, it appears from Fishman's work that noise measurements will probably be very useful in distinguishing between rival models of K⁺ channels.     

The Estimation of a Multivariate Fitness Function from Several Samples Taken from a Population

The situation is considered where the multivariate distribution of certain variables X₁, X₂, …, X_p is changing with time in a population because natural selection related to the X's is taking place. It is assumed that random samples taken from the population at times t₁, t₂, …, t_s are available and it is desirable to estimate the fitness function w_t(x₁, x₂,…,x_p) which shows how the number of individuals with X_i = x_i, i = 1, 2, …, p at time t is related to the number of individuals with the same X values at time zero. Tests for population changes are discussed and indices of the selection on the population dispersion and the population mean are proposed. The situation with a multivariate normal distribution is considered as a special case. A maximum likelihood method that can be applied with any form of population distribution is proposed for estimating w_t. The methods discussed in the paper are illustrated with data on four dimensions of male Egyptian skulls covering a time span from about 4500 B.C. to about 300 A.D. In this case there seems to have been very little selection on the population dispersion but considerable selection on means.     

Attempt at quantitative analysis of interindividual variability in stress response of leucocyte levels in mice from the viewpoint of control theory

Summary The authors studied the dependence between the ability of systems regulating the level of peripheral leucocytes to damp the fluctuations of this level, occurring under conditions of repeated stressing of the organism by non-specific stress factors (isolation, starvation, handling for blood withdrawal) and the tolerance of the organism to the action of ionizing radiation (summary exposure to X-rays leading to death at repeated irradiation). This regulatory ability was expressed with the aid of the slope (coefficientb) of regression between the leucocytary valuesx _i, fluctuating in time, and the subsequent differences _i ( _i =x _i+1–x _i ). The highest tolerance to lethal action of radiation is correlated to such a regulatory behaviour of the organism at which the time necessary for adaptation of the system is shortest. The authors point out the suitability of the employed regulatory characteristic for the quantitative study of adaptability and resistance of the organism and its relatedness to the damping ratio, used in the automatic control theory.     

The rate of convergence of a generalized stable population

In an age-structured population that grows exponentially, each age groupP _i(t) at periodt is asymptotically equivalent tox ₀ ^t for some positive number x₀. In this paper we show that the speed at which the ith age group reaches its exponential state of equilibrium can be measured by the rate at which the ratio v_i(t)=P_i(t)/p_i(t–1) converges tox ₀. The age specific rate of convergence is determined by considering a quantityr satisfyingv _i(t)-x ₀ ¦ r ^t whent is large;R _i=Infr (over all initial populations,r satisfying the above inequality) is the R-factor used in numerical analysis to measure the rate at which the sequencev _i (t) converges tox ₀;S _i =- In R_i is then defined as the rate of convergence to stability of the ith age group. The case of constant net maternity rates is studied in detail; in this contextS ₀ is compared to the population entropyH, which was proposed by Tuljapurkar (1982) as a measure of the rate of convergence to stability.     

The effect of toxins on inorganic phosphate release during actin polymerization

During the polymerization of actin, hydrolysis of bound ATP occurs in two consecutive steps: chemical cleavage of the high-energy nucleotide and slow release of the γ-phosphate. In this study the effect of phalloidin and jasplakinolide on the kinetics of P _i release was monitored during the formation of actin filaments. An enzyme-linked assay based spectrophotometric technique was used to follow the liberation of inorganic phosphate. It was verified that jasplakinolide reduced the P _i release in the same way as phalloidin. It was not possible to demonstrate long-range allosteric effects of the toxins by release of P _i from F-actin. The products of ATP hydrolysis were released by denaturation of the actin filaments. HPLC analysis of the samples revealed that the ATP in the toxin-bound region was completely hydrolysed into ADP and P _i . The effect of both toxins can be sufficiently explained by local and mechanical blockade of P _i dissociation.     

Characterization of the WEIBULL Distribution by Properties of Order Statistics

Let X_1:n, X_2:n, X_3:n…, X_n:n be the order statistics of n independent random variables with the common (absolutely continuous strictly increasing) distribution function F. The main results given in this article are:

• 1 For any fixed r and two distinct numbers s₁ and s₂ (1<r<s₁<s₂≦n) the distributions of V_i and W_i (defined in (1.11) and (1.12) are identical for i = 1,2 iff F(x) is WEIBULL (1.2).
• 2 The statistics D₁ and D₂ (as in (1.8) and (1.9)) are independent iff F(x) is WEIBULL (1.2).
• 3 The statistics U_i (1≦j≦n?1) and X_i:n (i ≦ j) are independent iff F(x) is WEIBULL (1.2).
• 4 Let X, X₁, X₂, …, X_k be random variables such that

These conditions are necessary and sufficient for F(x) to be WEIBULL .     

Role of ionization of the phosphate cosubstrate on phosphorolysis by purine nucleoside phosphorylase (PNP) of bacterial (<Emphasis Type="Italic">E. coli</Emphasis>) and mammalian (human) origin

Kinetics of the reactions of purine nucleoside phosphorylases (PNP) from E. coli (PNP-I, the product of the deoD gene) and human erythrocytes with their natural substrates guanosine (Guo), inosine (Ino), a substrate analogue N(7)-methylguanosine (m⁷Guo), and orthophosphate (P_i, natural cosubstrate) and its thiophosphate analogue (SP_i), found to be a weak cosubstrate, have been studied in the pH range 5–8. In this pH range Guo and Ino exist predominantly in the neutral forms (pK_a 9.2 and 8.8); m⁷Guo consists of an equilibrium mixture of the cationic and zwitterionic forms (pK_a 7.0); and P_i and SP_i exhibit equilibria between monoanionic and dianionic forms (pK_a 6.7 and 5.4, respectively). The phosphorolysis of m⁷Guo (at saturated concentration) with both enzymes exhibits Michaelis kinetics with SP_i, independently of pH. With P_i, the human enzyme shows Michaelis kinetics only at pH ∼5. However, in the pH range 5–8 for the bacterial enzyme, and 6–8 for the human enzyme, enzyme kinetics with P_i are best described by a model with high- and low-affinity states of the enzymes, denoted as enzyme-substrate complexes with one or two active sites occupied by P_i, characterized by two sets of enzyme-substrate dissociation constants (apparent Michaelis constants, K _m1 and K _m2) and apparent maximal velocities (V _max1 and V _max2). Their values, obtained from non-linear least-squares fittings of the Adair equation, were typical for negative cooperativity of both substrate binding (K _m1 < K _m2) and enzyme kinetics (V _max1/K _m1 > V _max2/K _m2). Comparison of the pH-dependence of the substrate properties of P_i versus SP_i points to both monoanionic and dianionic forms of P_i as substrates, with a marked preference for the dianionic species in the pH range 5–8, where the population of the P_i dianion varies from 2 to 95%, reflected by enzyme efficiency three orders of magnitude higher at pH 8 than that at pH 5. This is accompanied by an increase in negative cooperativity, characterized by a decrease in the Hill coefficient from n _H ∼1 to n _H ∼0.7 for Guo with the human enzyme, and to n _H ∼0.7 and 0.5 for m⁷Guo with the E. coli and human enzymes, respectively. Possible mechanisms of cooperativity are proposed. Attention is drawn to the substrate properties of SP_i in relation to its structure.     

Predictive models in aerobiology: data transformation

This paper attempts to evaluate the effect of mathematical transformations of pollen and meteorogical data used in aerobiological forecasting models. Stepwise multiple regression equations were developed in order to facilitate short term forecasts during the pre-peak period. The daily mean pollen data (x _i) expressed as number of pollen grains per cubic metre of air were used directly and transformed into different scales: log(x _i + 1), ln((x ₁1000/Σp) + 1) and √x _i, where Σp is the sum of the daily mean values throughout the season. Thirteen meteorological parameters and the variable time were used as forecasting variables. The most reliable forecasts were obtained with data transformed by ‘square root’ and with untransformed data. Based on the results obtained, we recommend that the data be transformed by means of the square root if they do not show a normal distribution and that non-linear statistics be used in this kind of study.     

On the Use of Ranks in Combined Estimates in Quantal Bioassays

If {U_i} i = 1, …, k is a sequence of binary responses of n_i subjects at each of k successive dose levels x_i, there is the problem of the statistical treatment of the observed proportions P_i = U_i/ni when neither the probit nor the logit transformation may be assumed. This paper considers the use of the midranks of the responses for point and interval estimation of relative potency in the case of parallel line assay in particular. More generally the problem of combining the results of several independent estimates using ranks is discussed. Several examples illustrate the method.     

ATP-dependent potassium channels of muscle cells: Their properties,regulation, and possible functions

ATP-dependent potassium channels are present at high density in the membranes of heart, skeletal, and smooth muscle and have a lowP _open at physiological [ATP]_i. The unitary conductance is 15–20 pS at physiological [K⁺] _o , and the channels are highly selective for K⁺. Certain sulfonylureas are specific blockers, and some K channel openers may also act through these channels. K_ATP channels are probably regulated through the binding of ATP, which may in turn be regulated through changes in the ADP/ATP ratio or in pH_i. There is some evidence for control through G-proteins. The channels have complex kinetics, with multiple open and closed states. The main effect of ATP is to increase occupancy of long-lived closed states. The channels may have a role in the control of excitability and probably act as a route for K⁺ loss from muscle during activity. In arterial smooth muscle they may act as targets for vasodilators.     

Effects of plant species on phosphorus availability in a range of grassland soils

Vegetative conversion from grass to forest may influence soil nutrient dynamics and availability. A short-term (40 weeks) glasshouse experiment was carried out to investigate the impacts of ryegrass (Lolium perenne) and radiata pine (Pinus radiata) on soil phosphorus (P) availability in 15 grassland soils collected across New Zealand using ³³P isotopic exchange kinetics (IEK) and chemical extraction methods. Results from this study showed that radiata pine took up more P (4.5–33.5 mg P pot^–1) than ryegrass (1.1–15.6 mg pot^–1) from the soil except in the Temuka soil in which the level of available P (e.g., E _1min P_i, bicarbonate extractable P_i) was very high. Radiata pine tended to be better able to access different forms of soil P, compared with ryegrass. There were no significant differences in the level of water soluble P (Cp, intensity factor) between soils under ryegrass and radiata pine, but the levels of Cp were generally lower compared with original soils due to plant uptake. The growth of both ryegrass and radiata pine resulted in the redistribution of soil P from the slowly exchangeable P_i pool (E _{> 10m} P_i, reduced by 31.8% on the average) to the rapidly exchangeable P_i (E _1min-1d P_i, E _1d-10m P_i) pools in most soils. The values of R/r ₁ (the capacity factor) were also generally greater in most soils under radiata pine compared with ryegrass. Specific P mineralisation rates were significantly greater for soils under radiata pine (8.4–21.9%) compared with ryegrass (0.5–10.8%), indicating that the growth of radiata pine enhanced mineralisation of soil organic P. This may partly be ascribed to greater root phosphatase activity for radiata pine than for ryegrass. Plant species × soil type interactions for most soil variables measured indicate that the impacts of plant species on soil P dynamics was strongly influenced by soil properties.     

Further Study on the Statistical Test to Detect Spatial Pattern

In the study of spatial patterns, the statistic I' = (n — 1)s²/x was commonly used. In this paper, we found that x — s² has an approximated normal distribution with zero mean if the x_i's (i = 1 to n) are independent identically distributed Poisson variables. Based on this conclusion, the hypothesis that a point pattern is completely random can be tested directly. And a method for the test of spatial patterns was proposed which can be sued as an alternative to the Chi-square based dispersion index test.     

Conformations and time-resolved fluorescence of oligomers of (−)-epicatechin with 4β → 8 interflavan bonds

Oligomers of (?)-epicatechin with 4β → 8 interflavan bonds, and as many as five monomer units, have been studied by conformational analysis and time-resolved fluorescence. The conformational analysis yields 2^x?1 conformations for each oligomer with x monomer units. There are two conformations accessible at each interflavan bond. These conformations are denoted by + and ?. The dominant conformations for the trimer and higher oligomers have an interaction between the phenolic hydroxyl groups on monomers i and i+2. This interaction involves the hydroxyl group at C(5) on monomer i, and either C(13) or C(8) of monomer i+2, depending on whether the conformation of the two intervening interflavan bonds is + + or + ?, respectively. Minor contributions to the ensemble for the tetramer and pentamer are made by conformations that contain the sequence of successive interflavan bonds denoted by ?+ or ??. In ?+ the interaction between monomer units i and i + 2 involves an aliphatic hydroxyl with a phenolic hydroxyl, and there are no hydroxyl-hydroxyl interactions between units iand i + 2 in ??. The onset of a different decay law for the fluorescence when x increases from 3 to 4 may be associated with the appearance of the ?+ and ?? conformations as minor constituents in the ensemble.     

Some Early Developments in Ratio Estimation

John Graunt (1662) was the first to estimate the ratio y/x where y represents the total population and x the known total number of registered births in the same areas during the preceding year. About 1765 Messance (Stephan, 1948) and Moheau (1778) published very carefully prepared estimates for France based on enumeration of population in certain districts and on the count of births, deaths and marriages as reported for the whole country. The districts from which the ratio of inhabitants to birth was determined only constituted a sample. Laplace (1786) prepared similar estimates in 1802 based on a two-stage sampling plan. Recently Hansen and Hurwitz (1943) showed that the ratio estimate (y_i/n_i)X of Y is unbiased where all x_i's are known and the n^th cluster is selected with p.p.s. More recently Hájek (1949), Lahiri (1951), Midzuno (1952) and Sen (1952) developed independently the sampling of n clusters with p.p.s to the totals of the sizes of the sample clusters S(x_i). Des Raj (1954) and Sen (1952, 1953) gave unbiased estimate of the variance of the estimator which was generally non-negative for samples with smaller probabilities. Rao and Vijayan (1977) gave an unbiased estimator which is non-negative for samples with larger probabilities. Hájek (1949) provided an almost unbiased estimator of the variance of the estimator. The paper discusses situations where Hájek's estimator of variance should be preferred to the Rao-Vijayan estimator and vice versa.     

Short communication: Purification and properties of acid phosphatase (EC 3.1.3.2) secreted by strain 74A of the mould Neurospora crassa

Both P_i-repressible acid phosphatases, IIb (mycelial) and IIc (extracellular), synthesized by Neurospora crassa and purified to apparent homogeneity by 7.5% PAGE, are monomers, are inhibited by 2 mm ZnCl₂ and are non-specifically stimulated by salts. However, the IIc form is activated by p-nitrophenylphosphate (in a negative co-operativity effect with a K _0.5 of 2.5 mm) whereas form IIb shows Michaelis kinetics, with a K _m of 0.5 mm. Thus, since both enzymatic forms may be expressed by the same gene (pho-3), it is possible that post-translational modifications lead to the excretion of an enzymatic form with altered Michaelis kinetics compared with the enzymatic form retained by the mycelium.