Foam behavior of biological media

Summary The surface tension and foaminess of (a) unlimited, (b) substrate limited, and (c) oxygen transfer limited growth media of Hansenula polymorpha were measured using methanol, ethanol or glucose as a substrate.The time dependence of can be described by the Avrami-Überreiter relationship: log (2.3 log V)=n log t+log b, where V = (_O – _eq/(_t – _eq, and _O, _t and _eq are at t_M=0, t_M=t and t_M (equilibrium value).The constants n and b are functions of the fermentation time t_F as long as the growth is unlimited but they are constant in the state of limited growth. With glucose substrate, the foaminess can be presented as a definite function of the time, t_DG, which is necessary to attain _eq. With alcohol as a substrate no definite (t_DG) function was found.Symbols b constant in Eq. (1) - n constant in Eq. (1) - S substrate concentration - T temperature - t_M time h (measured from the beginning of the determination of the surface tension ) - t_F cultivation time h (measured from the time of inoculation) - t_DG time (min) necessary to attain the equilibrium surface tension ) - X dry biomass concentration (gl^–1) - V (_O – _eq)/(_t – _eq) - V_S equilibrium volume of the foam (cm³) - V_G volumetric gas flow rate during the estimation of (cm³ s^–1) - vvm volumetric gas flow rate with regard to the volume of the medium (min^–1) - w_SG superficial gas velocity (cm s^–1) - _m maximum specific growth rate (h^–1) - V_S/V_G foaminess (s) - surface tension, mMm^–1 (milli Newton m^–1) - _O at t_M=0 - _eq equilibrium surface tension ( at t_M) - _t at t_M=t - HP probes from Hansenula polymorpha cultivation - NLG non limited growth - OTLG oxygen transfer limited growth - SLG substrate limited growth     

Observations on phase-locking within the response of primary muscle spindle afferents to pseudo-random stretch

In order to uncover encoder properties of primary muscle spindle afferent fibers, time coupling (phase-locking) of action potentials on cyclic muscle stretch was studied by means of pseudo-random noise. In cats Ia action potentials were recorded from dorsal root filaments and the gastrocnemius muscles of one hind leg were stretched. The stimulus time course was a determined sequence of randomly varying muscle length which could be applied repeatedly (sequence duration 0.6 or 20 s). The noise amplitude (standard deviation of displacements) was varied between 5 and 300 m, the upper cut-off frequency of noise f _c was varied between 20 and 100 Hz. The responses to the consecutive pseudo-random noise cycles were displayed as raster diagrams and cycle histograms. Phaselocking characterized the responses at all noise amplitudes outside the near threshold range (>10 m). The higher and f _c , the stronger was the phase-locking of impulses on the stretch. When and f _c were selected to achieve high mean stretch velocities of about 500 mm/s, phase-locking was as precise as 0.15 ms, measured as the variability of spike occurrences with respect to stretch. The rasters obtained with low noise amplitudes (<40 m) showed a loose phase-locking and this gave insight into underlying mechanisms: The elicitation of action potentials caused by dynamic stretch can be prevented by a post-spike depression of excitability. This disfacilitation was very effective in counteracting weak stretch components within the random sequence and less effective or even missing when relatively strong stretch components could force the spike elicitation. This led to the reestablishment of phase-locked patterns. The results were discussed in relation to the known encoder models.     

Electric field-induced breakdown of lipid bilayers and cell membranes: A thin viscoelastic film model

Summary A simple viscoelastic film model is presented, which predicts a breakdown electric potential having a dependence on the electric pulse length which approximates the available experimental data for the electric breakdown of lipid bilayers and cell membranes (summarized in the reviews of U. Zimmermann and J. Vienken, 1982,J. Membrane Biol. 67:165 and U. Zimmermann, 1982,Biochim. Biophys. Acta 694:227). The basic result is a formula for the time of membrane breakdown (up to the formation of pores): =(/C)/( _m ² ₀ ² U ⁴/24Gh ³+T ²/Gh–1), where is a proportionality coefficient approximately equal to ln(h/2₀),h being the membrane thickness and ₀ the amplitude of the initial membrane surface shape fluctuation ( is usually of the order of unity), represents the membrane shear viscosity,G the membranes shear elasticity modules, _m the membrane relative permittivity, ₀=8.85×10^–12 Fm,U the electric potential across the membrane, the membrane surface tension andT the membrane tension. This formula predicts a critical potentialU _c ;U _c =(24Gh ³/ _m ² ₀ ² )^1/4 (for = andT=0). It is proposed that the time course of the electric field-induced membrane breakdown can be divided into three stages: (i) growth of the membrane surface fluctuations, (ii) molecular rearrangements leading to membrane discontinuities, and (iii) expansion of the pores, resulting in the mechanical breakdown of the membrane.     

Theoretical studies on the necessary number of components in mixtures

Summary Theoretical studies on the optimal numbers of components in mixtures (for example multiclonal varieties or mixtures of lines) have been performed according to phenotypic yield stability (measured by the parameter variance). For each component i, i = 1, 2,..., n, a parameter u_i with 0 u_i 1 has been introduced reflecting the different survival and yielding ability of the components. For the stochastic analysis the mean of each u_i is denoted by u ₁ and its variance by _i ² For the character total yield the phenotypic variance V can be explicitly expressed dependent on 1) the number n of components in the mixture, 2) the mean of the _i ² 3) the variance of the _i ² 4) the ratio and 5) the ratio _i ² /² where denotes the mean of the u _i and _u ² is the variance of the u _j. According to the dependence of the phenotypic stability on these factors some conclusions can be easily derived from this V-formula. Furthermore, two different approaches for a calculation of necessary or optimal numbers of components using the phenotypic variance V are discussed: A. Determination of optimal numbers in the sense that a continued increase of the number of components brings about no further significant effect according to stability. B. A reduction of b % of the number of components but nevertheless an unchanged stability can be realized by an increase of the mean of the u _i by 1% (with and _u ² assumed to be unchanged). Numerical results on n (from A) and 1 (from B) are given. Computing the coefficient of variation v for the character total yield and solving for the number n of components one obtains an explicit expression for n dependent on v and the factors 2.-5. mentioned above. In the special case of equal variances, _i ² = _o ² for each i, the number n depends on v, x = (₀/)² and y = (_u/)². Detailed numerical results for n = n (v, x, y) are given. For x 1 and y 1 one obtains n = 9, 20 and 79 for v = 0.30, 0.20 and 0.10, respectively while for x 1 and arbitrary y-values the results are n = 11, 24 and 95.This publication is an extended version of a lecture given at the 1984-EUCARPIA meeting (Section Biometrics in Plant Breeding) in Stuttgart-Hohenheim (Federal Republic of Germany)     

Amber mutations in the structural gene for RNA polymerase sigma factor of Escherichia coli

Summary Amber mutants of Escherichia coli K-12 affected in the structural gene (rpoD) for th subunit of RNA polymerase have been obtained from a strain harboring a temperature-sensitive amber suppressor (supF-Ts6) which is active only at low temperatures. These mutants grow normally at low temperature (30°C) but do not grow at high temperature (42°C) due to the inability to synthesize factor. In one mutant studied in detail (rpoD40), the rate of -factor synthesis at 30°C is about half that of the wild type and is decreased to 10%–15% within 1 h of incubation at 42°C. The synthesis of core polymerase subunits or bulk protein is virtually unaffected at least for 2 h. The defect of the mutant in synthesis and growth at high temperature can be suppressed by any of the amber suppressors tested (supD, supE or supF). RNA-polymerase holoenzymes prepared from the mutant cells carrying each of the suppressors (grown at 42°C) exhibit different thermostabilities attributable to alterations in the factor. The reduced synthesis in the mutant is accompanied by the synthesis of polypeptide tentatively identified as amber fragment. These results as well as the genetic mapping data indicate that the amber mutation (rpoD40) resides within the structural gene for the factor and directly affects synthesis upon inactivation of the suppressor at high temperature.     

A method for the estimation of gene flow parameters from a population structure caused by restricted gene flow and genetic drift

Summary A method has been developed which enables the estimation of the plant gene flow parameters _p (pollen dispersal), _s (seed dispersal) and t (outcrossing rate) from a selection-free continuously structured population in equilibrium. The method uses Wright's F-coefficients and introduces a new F-function which describes the genetic similarity as a function of the spatial distance. The method has been elaborated for wind pollinated plant species but can be modified for insect pollination and for animal species. In practice allozymes will provide for the necessary neutral genetic variation. The more loci used and the more intermediate the gene frequencies, the more reliable the results. For the estimation of _p and t together (when the outcrossing rate is not known) at least two chromosomally unlinked loci are required. The method for estimating _s depends on whether the plant species is annual or perennial. The mechanism of selfing has been analysed by the explanation of the value of t by three components: population density (d), pollen flow (_p) and relative fertilization potential of own pollen (Z). The concepts of neighbourhood size and isolation by distance, developed by Wright, who used a single gene flow parameter , have been extended to the situation which is realistic for seed plants, using all three parameters _p, _s and t. When _p is large with respect to _s, _s largely determines the value of the neighbourhood size, whereas _p is the most dominating factor in isolation by distance. The use of local effective population size and mean gene transport per generation instead of neighbourhood size and neighbourhood area, respectively, is proposed to avoid confusion. Computer simulations have been carried out to check the validity and the reliability of the method. Populations of 200 plants, using two or three loci with intermediate allele frequencies, gave good results in the calculation of _p with known value of t and of _s and N_e. With unknown t, especially with lower values of t, larger populations of at least 1,000 plants are necessary to obtain reasonably accurate results for _p and mean gene transport per generation M.Grassland Species Research Group Publication No. 81     

Characterizing the symmetric equilibrium of multi-strain host-pathogen systems in the presence of cross immunity

We investigate the population dynamics of host-pathogen systems in which the pathogen has a potentially arbitrary number of antigenically distinct strains interacting via cross-immunity. The interior equilibrium configuration of the symmetric multiple strain SIR model with cross-immunity is characterized. We develop an efficient iterative method for numerically solving the equilibrium equation together with a number of informative analytical approximations to the full solution. Equilibrium properties are studied as a function of the number of strains, reproduction number, infectious period, and cross immunity profile. We establish that the prevalence in the system increases monotonically with the number of strains and the reduction in cross immunity. Moreover, we demonstrate the existence of a phase transition separating high prevalence and low prevalence parameter regions, with the critical point being defined by R₀1, where is the level of cross-immunity and R₀ is the reproduction number. Above the threshold, prevalence saturates with increasing numbers of strains as a result of the inclusion of prohibition of co-infection in the model. Below the threshold, prevalence saturates much more rapidly as the number of strains increases - indicating that when cross-protection is sufficiently intense, the selective advantage for a pathogen to increase its diversity is substantially less than in the threshold region. Similarly, there is limited benefit to increased transmissibility (or decreased cross-immunity) both for the high and low diversity pathogen systems compared with systems at the threshold R₀1 where small increase in transmissibility can result in significant increase in prevalence.     

RNA polymerase sigma-related proteins in Escherichia coli: detection by antibodies against a synthetic peptide

Summary Antibodies were raised against a synthetic tetradecameric peptide with an amino acid sequence, DLIQEGNIGLMKAV, which corresponds to the most highly conserved region of bacterial RNA polymerase factors. In a Western-blot analysis of total Escherichia coli proteins, the antiserum reacted specifically with at least three proteins with apparent molecular weights of 75 kDa, 27 kDa and 23 kDa, in addition to the known factors (⁷⁰ and ³²). The majorities of ⁷⁰ and ³² were recovered as associated forms with the RNA polymerase on glycerol gradient centrifugation, while the other cross-reacting proteins were not. Unambiguous evidence was obtained which indicated that the intracellular level of ³² increased rapidly upon heatshock, at least in the strain containing high copy numbers of the rpoH gene.     

Intracellular responses from the photosensitive pineal organ of the teleost,Phoxinus phoxinus

Summary Intracellular potentials from the isolated dark-adapted pineal organ ofPhoxinus phoxinus were recorded by using glass microelectrodes. The majority of cells had resting potentials of 20 to 35 mV and responded to light with intensitygraded hyperpolarizations. Voltage intensity curves of responses to brief flashes followed the hyperbolic tangent functionV/V _max=Iⁿ/(I ⁿ + ⁿ ).The latency of onset for responses to light stimuli near threshold was 400 ms and decreased with saturating flashes to about 50 ms. The membrane resistance decreased during the hyperpolarization. Spectral sensitivity measurements for these cells exhibited curves with _max=530 nm. Intracellular dye injection unequivocally identified this cell type as a photoreceptor cell.A second cell type with resting potentials between 30 to 40 mV exhibited a biphasic response pattern to light stimulation. The cell depolarized with dim light flashes and hyperpolarized with bright flashes. The amplitude of the hyperpolarizing component showed no saturation over an intensity range of 5 log units. Latencies and rise times were comparable to those of photoreceptor potentials. Spectral sensitivity curves peaked at longer wavelengths ( _max=550 nm) than the action spectra of photoreceptors ( _max=530 nm). It is assumed that this rare cell type represents a small class of pineal interneurons.     

Properties of components of covariance of inbred relatives and their estimates in a maize population

Summary Properties of three parameterizations, denoted as the C-model, D-model and Q-model, for covariances of inbred relatives under assumptions of no linkage or epistasis are explored and compared. Additive variance in an inbred population with inbreeding coefficient F, ² _AF =(1+F) ² _A where ² _A is additive variance in a panmictic population, if Q-model parameters Q _xx and Q _xy are both zero. Conditions sufficient for this to hold are presented in terms of gene frequencies and dominance contrasts (homozygotes vs. heterozygotes). Some other properties and potential uses of estimates of components in the models are also discussed. Estimates of components in the D-model and Q-model were calculated from a maize (Zea mays L.) study from which estimates of components in the C-model were previously published. Of particular interest were the covariance (Q _xy ) of effects of alleles at complete homozygosity with inbreeding depression effects, the covariance (D ₁) of additive effects at panmixia with inbreeding depression effects and the within-locus variance (D ₂, alias Q _xx ) of inbreeding depression effects. Estimates of Q _xy , D ₁, and D ₂ were small and nonsignificant in most cases. For ear height in the second year of the study, D ₂ appeared to be a major component. In some cases, results were obtained which had contradictory implications (negative D ₂ coupled with positive Q _xy or D ₁, and positive D ₂ coupled with negative ² _D ). A negative estimate of one or the other of ² _D or ² _A was obtained in one of the two within-year analyses for every character. Problems in getting realistic results were thought to be owing to excessive multicollinearity among the coefficients of the components in the expectations of the covariances of the kinds of relatives included in the study. Implications for future studies of this kind are discussed.Journal Article No. 87-3-14 of the Kentucky Agricultural Experiment Station published with the approval of the Director     

A comparative study of variation in codon 33 of the<Emphasis Type="Italic"> rpoS</Emphasis> gene in<Emphasis Type="Italic"> Escherichia coli</Emphasis> K12 stocks: implications for the synthesis of σ<Superscript>s</Superscript>

The Escherichia coli rpoS gene encodes an RNA polymerase sigma factor (sigma S or ^S) required for the expression of stationary-phase genes. In the first published rpoS sequence from E. coli K-12 codon 33 is given as CAG. However, several subsequent independent studies found the amber codon TAG at this position ( rpoSAm). Besides this amber codon, other codons such as TAT have also been found at this location in rpoS. Comparative genome analysis now leads us to propose TAG as the parental codon 33 in rpoS in E. coli K-12. Five different stocks of the strain W3110, which differ in the levels of ^S protein they express, were investigated. We sequenced the rpoS gene from these, and found a T at nucleotide position 97 in four out of the five stocks and a G at position 99 in three out of the five. W1485, a parental strain of W3110, and W3350, a derivative of W3110, are also rpoSAm mutants. Such rpoSAm mutants would be expected to show no RpoS activity. The retention of partial or intermediate ^S activity by suppressor-free rpoSAm mutants is therefore puzzling. We propose that a functional, N-terminally truncated, ^S (1–53^S) can be translated from a Secondary Translation Initiation Region (STIR) located downstream of the amber codon 33. It has recently been reported that a fragment of RpoS (1–53^S) that lacks the first 53 amino acids is functional when synthesized in vivo. Taken together, our results support the hypothesis that the original codon 33 of the rpoS gene in E. coli K-12 strains is the amber codon TAG.Communicated by W. Goebel     

Studies of inheritance of reaction to common smut in corn

Summary Reaction to Ustilago maydis was studied in resistant and susceptible corn inbreds, their F ₁ hybrids and F ₂ and F ₃ segregants. Marked differences among inbreds in genetic prepotency were found. Segregation was polygenic. The concept of combining ability was applied and estimates of _G ^/2 and _S ^/2 were calculated. Both additive and non-additive gene action was found. Breeding for resistance based on crossing to special susceptible testers was suggested.

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Zusammenfassung An resistenten und anfälligen Mais-Inzuchtlinien, deren F ₁-Hybriden und deren F ₂- und F ₃- Generationen wurde die Reaktion auf Ustilago maydis untersucht. Bezüglich der genetischen Präpotenz zeigten sich bei den Inzuchtlinien deutliche Unterschiede. Die Spaltung war polygen. Es wurde die Methode der Prüfung der Kombinationsfähigkeit angewandt und die allgemeine ( _G ² ) und spezielle ( _G ^/2 ) Kombinationsfähigkeit geschätzt. Sowohl additive wie nicht-additive Genwirkung wurde gefunden. Für die Resistenzzüchtung werden Kreuzungen mit besonders anfälligen Testern vorgeschlagen.

     

Rates of decay for the survival probability of a mutant gene II The multitype case

This paper extends the results of [1] to the multitype case. For a multitype branching process that is slightly supercritical, approximations for the survival probability in terms of the maximal eigenvalue of the mean matrix and a generalized variance ² are developed. Our results improve upon those of Hoppe [5] and Eshel [3] that seek to validate a conjecture of Ewens [4].Research supported in part by NSF grant DMS 9007182     

Efficiency of check-plot designs in unreplicated field trials

Check-plot designs have a lower selection intensity than unreplicated non-check-plot designs if both the number of test lines to be selected (s) and of total plots in the trial (N) are kept constant. For a check-plot design to be more efficient, local control must effectively reduce the plot error variance and increase heritability to such a level that it compensates for the corresponding loss in selection intensity and makes the expected gain from selection at least equal to that in the non-check-plot design. To realize this goal, the required minimum reduction in plot error variance in a checkplot design (relative to that in a non-check-plot design) depends on (1) check-plot frequencyf _c , (2) fractionk = s/N, and (3) ratiow ₀= ₀ ² / _g ² of non-check-plot design plot error variance ₀ ² to genetic variance _g ² among test lines. Lowerw ₀ and higherf _c andk are found to require a relatively higher reduction in plot error variance in check-plot designs. A condition is derived to show when a check-plot design may never be more efficient.     

Potential induced in a spherical cell by an intracellular point source and an extracellular point sink

Summary The potential is calculated for all time, inside and outside a spherical cell for a point source of current inside the cell and a point sink located a finite distance outside the cell. The source and sink are step functions in time. An eigenfunction expansion is obtained, valid for arbitrary =_m a/_i , where _i and _m are the conductivities inside the cell and in the membrane, respectively, a is the cell radius and the membrane thickness. For small , the eigenfunction expansion is expanded in powers of . The time dependence of the potential contains transients with two widely differing time constants =C_m a/_i, where C_m is the membrane surface capacitance, and _m=/. Closed-form expressions are obtained for the two leading terms, for small , after the rapid transient is over. The remaining time dependence is only in the potential inside the cell, and is a simple exponential increase, independent of position within the cell. It is found that the transmembrane potential is insensitive to the location of the extracellular sink at long times, but not at short times. The dependence of the potential on location of source, sink, and observer is studied for long times after the quick transients are over. A uniqueness theorem is derived for the solution to Laplace's equation for the membrane boundary condition.This work was supported in part by NSF Grant No. GB-24965. Dr. Peskoff is the recipient of NIH Special Research Fellowship No. 1F03 GM 55849-01. Mr. Ramirez is a Ford Foundation Pre-doctoral Fellow.     

Dielectric properties of human blood and erythrocytes at radio frequencies (0.2–10 MHz); dependence on cell volume fraction and medium composition

The dielectric properties of human erythrocytes (red blood cells) suspended in whole blood and in isotonic media at various volume fractions (haematocrits) have been studied in the frequency range 0.2–10 MHz, in which the so-called-dispersion due to the Maxwell-Wagner effect is known to occur. The capacitance and conductance at 25 °C were measured by an instrument interfaced to a computer. The rectangular sample cavity (1 ml volume) contained four pure gold electrode pins, and the sample could be circulated by a roller pump. The frequency-dependence of the permittivity and conductivity were fitted by non-linear least squares regression. Corrections were applied for non-linearity in the dielectric increment at high haematocrit, and for electrode polarisation when diluting the blood in saline. Data were interpreted in terms of a simple equivalent resistor-capacitor circuit. From the measured haematological values the specific membrane capacitance (C_m) and the conductivities internal and external to the cells (_i and _o respectively) were estimated. The conductivities behaved in a predictable manner with a mean of 0.458 S · m^–1 (s.d. ± 0.044) for _i, whereas the value of C_m (and indeed the actual capacitance of the suspension) was dependent on the amount of plasma present. Hence, in stationary normal (anticoagulated) whole blood samples, C_m was as high as 2.98 F · cm^–2 (s.d. ± 0.40), in contrast to about 0.9 F · cm^–2 in blood diluted more than two-fold (to less than 20% hct) in isotonic media. The high value remained when the diluent was plasma. The C_m value returned to a high value when washed erythrocytes were reconstituted with plasma, provided that this was present at above a critical or threshold concentration of about 30 vol % in the medium, irrespective of the haematocrit in the range studied (15–44%). The C_m remained low in serum. When added to washed cells in saline, purified fibrinogen had no effect. However, high C_m values were obtained by fibrinogen supplementation to serum and diluted plasma. Applying moderate flow to whole blood approximately halved its high C_m value in an exponential manner with flow rate, whilst the C_m of washed cells (31–67% hct) slightly increased, and converged to the value for whole blood under flow. We interpret the highapparent C_m value in stationary samples to be a result of rapid cell aggregation in the presence of plasma, where rouleaux formation takes place before visible sedimentation sets in.     

Application of a stability statistic to international maize yield trials

Summary Genotype x environment (GE) interaction encountered in experiments complicates genotype selection and varietal recommendation. The integration of yield and stability of genotypes into a single parameter may make selection and recommendation easier. Kang developed a rank-sum method that allows selection for both yield and the stability variance statistics ( _i ² or s _i ² ) of Shukla. The objective of this research was to compare the rank-sum selection method to selection based on yield alone in five international maize (Zea mays L.) yield trials. Ranks were assigned for yield (the highest mean yield received a rank of 1) and for _i ² and s _i ² (the lowest value received a rank of 1). The yield and _i ² ranks and/or the yield and s _i ² ranks for each genotype were summed. Each trial contained two reference entries (REs). Yield rank or rank-sum of each genotype was compared to yield rank or rank-sum of the best RE (BRE). GE interaction was significant for all trials. Heterogeneity in the GE interaction due to the linear effect of a covariate (differences in fertility and/or cultural practices) was significant in Trials 1, 2, and 5. Overall, in all trials, 29 genotypes were selected on the basis of yield alone. On the basis of _i ² and yield rank-sum, 32 genotypes were identified, with 11 being lower yielding than the 29 yield-based selections. On the basis of s _i ² and yield rank-sum, 31 genotypes were selected, with 11 being lower yielding than the yield-bases selections. Obviously, yield is sacrificed when the rank-sum method is used in the selection process. However, selection based on yield alone may not be adequate when GE interaction is significant because of testing in diverse environments.     

Genetic architecture of a heterotic cross between two populations of maize

Summary The nature and magnitude of variability in the interpopulation cross of Mezcla Amarillo Selection (MAS), an introduction from CIMMYT, Mexico, and J607, a population developed in India using indigenous, American, and Yugoslavian germplasm, were studied. Interpopulation progenies developed by following the North Carolina Design I were evaluated at two locations. The additive genetic variance component in interpopulation cross, _A(12) ² , and in one population assuming the other population as tester, _A12 ² and _A21 ² were significant for all the traits evaluated, namely ear length, ear girth, kernel rows and days to silk, with one exception. For kernel rows, the dominance variance component, _A(12) ² , was also significant but it was smaller than _A(12) ² . The variance component due to dominance X location interaction, _DL(12) ² , was significant for all traits except kernel rows. In the case of ear length and ear girth, _DL(12) ² was greater than the other components. _AL(12) ² , _AL12 ² and _AL21 ² were not significant for any trait. Expected genetic advance indicated a superiority of half-sib reciprocal recurrent selection over full-sib reciprocal recurrent selection.     

Simple diagnostic statistical tests of models for DNA substitution

The accuracy of models for DNA substitution used in phylogenetic analyses is becoming more important with the increasing availability and analysis of molecular sequence data. It is natural to look for ways of improving these models, and to do this in a planned manner it is useful to be able to identify features of sequences that may not be described adequately. In this paper, I describe three statistics which may give useful diagnostic information on departures from models' predictions. The statistical distributions of these statistics are discussed and simple significance tests are derived. These tests are based on the (estimated) phylogeny of the sequences and so have the advantage of using the information contained in this tree. Examples are given of the application of the new tests to Markov chain models describing the evolution of primate pseudogene sequences and small-subunit RNA sequences.Abbreviations b(N,p) binomial distribution of N trials, each with probability p of success - m(N,p ₁,p ₂, ..., p _r ) multinomial distribution of N trials, with r possible outcomes having probabilities p ₁, p ₂, ..., p_r, respectively - N(, ²) Normal distribution with mean and variance ² - p() Poisson distribution with mean - bp base pairs - cdf cumulative distribution function - i.i.d. independent, identical distribution     

Relationships between genotype x environment interactions and rank orders for a set of genotypes tested in different environments

Multilocation trials in plant breeding lead to cross-classified data sets with rows=genotypes and columns=environments, where the breeder is particularly interested in the rank orders of the genotypes in the different environments. Non-identical rank orders are the result of genotype x environment interactions. Not every interaction, however, causes rank changes among the genotypes (rank-interaction). From a breeder's point of view, interaction is tolerable only as long as it does not affect the rank orders. Therefore, the question arises of under which circumstances does interaction become rank-interaction. This paper contributes to our understanding of this topic. In our study we emphasized the detection of relationships between the similarity of the rank orders (measured by Kendall's coefficient of concordance W) and the functions of the diverse variance components (genotypes, environments, interaction, error). On the basis of extensive data sets on different agricultural crops (faba bean, fodder beet, sugar beet, oats, winter rape) obtained from registration trials (1985–1989) carried out in the Federal Republic of Germany, we obtained the following as main result: W ₂ ^g /( ₂ ^g + ₂ ^v ) where ₂ ^g =genotypic variance and ₂ ^v = ₂ ^ge + ₂ ^o /L with ₂ ^ge =interaction variance, ₂ ^o =error variance and L=number of replications.