Transient and steady state phase response curves of limit cycle oscillators

The experiment of phase shifts resulting from discrete perturbations of stable biological rhythms has been carried out to study entrainment behavior of oscillators. There are two kinds of phase response curves, which are measured in experiments, according to as one measures the phase shifts immediately or long after the perturbation. The former is the first transient phase response curve and the latter is the steady state phase response curve. We redefine both curves within the framework of dynamical system theory and homotopy theory. Several topological properties of both curves are clarified. Consequently, it is shown that we must compare the shapes of both two phase response curves to investigate the inner structures of biological oscillators. Moreover, we prove that a single limit cycle oscillator involving only two variables cannot simulate transient resetting behavior reported by Pittendrigh and Minis (1964). In other words, the circadian oscillator of Drosophila pseudoobscura does not consist of a single oscillator of two variables. Finally we show that a model which consists of two limit cycle oscillators is able to simulate qualitatively the phase response curves of Drosophila.     

Modelling the simultaneous processes occurring during the BOD5 test using synthetic waters as a pattern

Synthetic waters imitating BOD₅ test conditions have been studied to model the progress curves of cell growth, organic matter consumption and oxygen demand. Glucose and glutamic acid were used as organic matter and Bacillus subtilis, Escherichia coli and Pseudomonas putida as microorganisms. The concentrations of all the products were measured simultaneously over time and were fitted to the Logistic and Monod models. Following this, the goodness of fit was analysed. Discrimination between models allowed us to conclude that the Logistic model has acceptable accuracy to describe the kinetic behaviour inside a BOD₅ bottle, while the extra parameter in the Monod model would not be statistically justified. Thus, the logistic parameters were determined under different conditions by non-linear regression and their biological meaning was interpreted.     

Biosorption Processes for Natural and Waste Water Treatment – Part II: Experimental Studies and Theoretical Model of a Biosorption Fixed Bed

A model has been developed for fixed‐bed biosorption performance, i.e. combined action of adsorption of organic water contaminants and their biological destruction in a column. The model contains an adsorption isotherm of the Freundlich type, adsorption kinetics by an overall film mass transfer (Glueckauf equation), maximum bacterial growth,and biological aerobic destruction (Monod model) of the organics by exoenzymes. Bacteria can not penetrate into the pores of the adsorbent. The model was tested using the system aqueous solution of aniline/Pseudomonas putida/Polysorb 40/100. Breakthrough curves in shorter columns have been measured and a velocity‐dependent steady‐state exit concentration was achieved. These curves could be simulated with sufficient accuracy on the basis of isotherm data, mass transfer coefficients and values of biological growth and destruction activity estimated from independent measurements.     

Determination of the antenna heterogeneity of Photosystem II by direct simultaneous fitting of several fluorescence rise curves measured with DCMU at different light intensities

Several methods for determination of the antenna heterogeneity of Photosystem II from fluorescence rise curves measured with DCMU have been developed so far. Using these methods, two, three or four types of Photosystem II with respect to the antenna heterogeneity were determined. However, the accuracy of some of these methods is under debate. Here, we present a new method for the determination of the antenna heterogeneity of Photosystem II. The method is based on direct simultaneous fitting of several fluorescence rise curves measured with DCMU at different intensities of light excitation. As several curves measured under different light conditions are fitted simultaneously by the same model, reliability and accuracy in determination of model parameters increase. Our method was applied to two plant materials with different structure of the thylakoid membrane: wheat leaves and cells of green alga Chlamydomonas reinhardtii. This revised version was published online in June 2006 with corrections to the Cover Date.     

Validation of fractal-like kinetic models by time-resolved binding kinetics of dansylamide and carbonic anhydrase in crowded media

Kinetic studies of biochemical reactions are typically carried out in a dilute solution that rarely contains anything more than reactants, products, and buffers. In such studies, mass-action-based kinetic models are used to analyze the progress curves. However, intracellular compartments are crowded by macromolecules. Therefore, we investigated the adequacy of the proposed generalizations of the mass-action model, which are meant to describe reactions in crowded media. To validate these models, we measured time-resolved kinetics for dansylamide binding to carbonic anhydrase in solutions crowded with polyethylene glycol and Ficoll. The measured progress curves clearly show the effects of crowding. The fractal-like model proposed by Savageau was used to fit these curves. In this model, the association rate coefficient k_a allometrically depends on concentrations of reactants. We also considered the fractal kinetic model proposed by Schnell and Turner, in which k_a depends on time according to a Zipf-Mandelbrot distribution, and some generalizations of these models. We found that the generalization of the mass-action model, in which association and dissociation rate coefficients are concentration-dependent, represents the preferred model. Other models based on time-dependent rate coefficients were inadequate or not preferred by model selection criteria.     

Theoretical calculations of the helix–coil transition of DNA in the presence of large,cooperatively binding ligands

Theoretical calculations are conducted on the helix–coil transition of DNA, in the presence of large, cooperatively binding ligands modeled after the DNA-binding proteins of current biological interest. The ligands are allowed to bind both to helx and to coil, to cover up any number of bases or base pairs in the complex, and to interact cooperatively with their nearest neighbors. The DNA is treated in the infinite homogeneous Ising model approximation, and all calculations are done by Lifson's method of sequence-generating functions. DNA melting curves are calculated by computer in order to expolore the effects on the transition of ligand size, binding constant, free activity, and ligand–ligand cooperativity. The calculations indicate that (1) at the same intrinsic free energy change per base pair of the complexes, small ligands, for purely entropic reasons, are more effective than are large ligands in shifting the DNA melting temperature; (2) the response of the DNA melting temperature to increased ligand binding constant K and/or free ligand activity L is adequately represented at high values of KL (but not at low KL) by a simple independent site model; (3) if curves are calculated with the total amount of added ligand remaining constant and the free ligand activity allowed to vary throughout the transition, biphasic melting curves can be obtained in the complete absence of ligand–ligand cooperativity. In an Appendix, the denaturation of poly[d(A-T)] in the presence of the drug, netropsin, is used to verify some features of the theory and to illustrate how the theory can be used to obtain numerical estimates of the ligand binding parameters from the experimental melting curves.     

Can telomere shortening explain sigmoidal growth curves?

A general branching process model is proposed to describe the shortening of telomeres in eukaryotic chromosomes. The model is flexible and incorporates many special cases to be found in the literature. In particular, we show how telomere shortening can give rise to sigmoidal growth curves, an idea first expressed by Portugal et al. [A computational model for telomere-dependent cell-replicative aging, BioSystems 91 (2008), pp. 262–267]. We also demonstrate how other types of growth curves arise if telomere shortening is mitigated by other cellular processes. We compare our results with published data sets from the biological literature.     

Electrophoresis of colloidal biological particles

Microscope electrophoresis was used to measure the electrophoretic mobility of polystyrene latex particles and bacterial, and mammalian tissue cells. The submicroscopic hydrophilic colloids (gelatin, serum albumin, and staphylococcal enterotoxin B) were adsorbed on latex carrier particles to determine their electrophoretic mobility and the effect of concentration, pH, electrolyte addition, and buffer ionic strength. Mobility curves as a function of pH were established for latex particles at 1 ppm concentration indicating an isoelectric point (IEP) at pH 3.6. The IEP for Escherichia coli B cells was measured at pH 2.8, Serratia marcescens at pH 2.6, Bacillus subtilis var. niger at pH 2.9, and L strain mouse fibroblast cells at pH 4.4. Using an adsorption technique, isoelectric points were measured for proteins: gelatin (acid form) at pH 9.4, serum albumin at pH 4.9, and staphylococcal enterotoxin B at pH 6.3. Procefures for examining electrophoretic characteristics of microscopic and submicroscopic biological particles are described in order to standardize procedures and to generate results applicable to an understanding of parameters influencing concentration and purification of colloidal biological particles.     

Elements pour une théorie statistique du tableau phytosociologique homogène

Summary A new model, based on information theory, for determining the equation of Raunkiaer frequency curves in a homogeneous phytosociological table is proposed. The forms of these curves are explained in relation to the geographical area covered by the relevés of the phytosociological table involved. The statistical distribution of species in a syntaxon (and the variability of a syntaxon as measured, by (E/E _m)-1, where E is the total number of species in the table, and E _m is the average number of species per relevé) are closely correlated with the size of the distribution area of relevés. Many examples are given and several consequences for the phytosociological method are indicated.

     

Fractals and the analysis of growth paths

A simple practical method exists for classifying and comparing planar curves composed of connected line segments. This method assigns, a single numberD, the fractal dimension, to each curve.D=log(n)/[log(n)+log(d/L)], where:n is the number of line segments,L is the total length of the line segments, andd is the planar diameter of the curve (the greatest distance between any two endpoints). At one end of the spectrum, for straight line curves,D=1; at the other end of the spectrum, for random walk curves,D→2. Standard statistics are done on the logarithms of the fractal dimension [log(D)]. With this measure, trails of biological movement, such as the growth paths of the cells and the paths of wandering organisms, can be analyzed to determine the likelihood that these trails are random walks and also to compare the straightness of the trails before and after experimental interventions.     

An equilibrium folding intermediate detected in the thermal unfolding transition of ribonuclease S by circular dichroism

The thermal-denaturation transition of ribonuclease S (RNAase S) is measured by circular dichroism at 225 nm. Only conformational transitions involving the S-peptide–S-protein complex are detected at this wavelength. Different pathways of thermal unfolding at high and low concentrations are apparent: at low concentrations the temperature of half-completion of denaturation (T_m) varies with concentration. Above a total enzyme concentration of 50 μM, T_m remains constant. The observed data can be explained on the basis of a model where the association–dissociation step occurs between S-peptide and thermally (at least partly) unfolded S-protein. The complex as a whole undergoes a major folding–unfolding transition in the course of which the S-peptide μ-helix appears to be formed. The unfolded complex is well populated in the unfolding transition region for enzyme concentrations of 100 μM or more. The model succeeds in deducing thermodynamic parameters from the thermal denaturation curves in various different ways. The values thus obtained are fully self-consistent and, moreover, consistent with the values for the apparent association constant and apparent association enthalpy as measured in enzyme-dilution experiments and by batch calorimetry.     

A mathematical model for describing light-response curves in <Emphasis Type="Italic">Nicotiana tabacum</Emphasis> L.

A modified exponential model was used to describe light-response curves of Nicotiana tabacum L. The accuracies of an exponential model, a nonrectangular hyperbola model, a rectangular hyperbola model, a modified rectangular hyperbola model and the modified exponential model were evaluated by Mean square error (MSE) and Mean absolute error (MAE). The tests MSE and MAE of the modified exponential model were the lowest among the five models. The light saturation point (LSP) obtained by the exponential model, the nonrectangular hyperbola model and the rectangular hyperbola model were much lower than the measured values, and the maximum net photosynthetic rates (P _max) calculated from these models, were greater than the measured values. P _max at LSP of 1,077 μmol m⁻² s⁻¹ calculated by the modified exponential model was 12.34 μmol(CO₂) m⁻² s⁻¹, which was more accurate than the values obtained from the modified rectangular hyperbola model. The results show that the modified exponential model is superior to other models for describing light-response curves.     

Analysis of Colony-Forming Ability of Human Fibroblast Strains by Linear Regression

A linear regression approach is presented for the statistical analysis of dose-response curves obtained by measuring the colony-forming ability of human fibroblast strains. The crucial determination of the dose range in which the linear model can be assumed is achieved by a combination of statistical criteria and biological claims. As a basic quantitative parameter we investigate the slope of the regression line and, by taking reciprocals, we retransform it into the biologically established parameter D₀. Several methods for the combination of estimates are presented.     

Evaluation of the potential to measure photosynthetic rates in C3 plants (Flaveria pringlei and Oryza sativa) by combining chlorophyll fluorescence analysis and a stomatal conductance model

The response curves of leaf photosynthesis to varying light, temperature and leaf-to-air vapour pressure deficit were measured in the C₃ plants Flaveria pringlei and Oryza sativa in normal air with a computerized open infrared gas analysis (IRGA) system, and the photochemical efficiency of photosystem II, described as (1–F,/F′_m) after Genty. Briantais & Baker (1989, Biochimica et Biophysica Acta 990, 87–92), was simultaneously measured with a modulated fluorometer. A model was written for rates of CO₂ fixation as a function of the true rate of O₂ evolution measured by fluorescene analysis (Jo₂), mesophyll conductance and intercellular CO₂ partial pressure. A second model was developed for rates of CO₂ fixation as a function of Jo₂, mesophyll conductance and stomatal conductance. In the latter case, leaf stomatal conductance was simulated using the stomatal model proposed by Leuning (1995, Plant, Cell and Environment 18 , 339–355). The rates of CO₂ fixation predicted from the models were similar to rates measured by IRGA. The results indicate that there is potential to measure CO₂ fixation in C₃ plants by combining the non-invasive measurement of Jo₂ by chlorophyll fluorescence analysis with the stomatal conductance model.     

A generalization of the clonal survival models: Equations for the families of curves obtained with fractionated irradiation

Summary The survival curves obtained when cellular recovery follows various first radiation dose deliveriesDI seem, when semi-logarithmically plotted, to be translated from the part of the curve corresponding to an unfractionated irradiation beyond a doseDR. A possible assumption consistent with such experimental observations is proposed which allows the generalization of any survival modelS = f(D). The derived equationS = f(DR + D - DI)f(DI)/f(DR) is convenient for the whole family of experimental survival curves involving cellular damage repairs when the first radiation doses vary. All the parameters of the family equation can be simultaneously fitted so that their reliability is increased. The generalized equations are given for the four following models: two-hits targets, Chadwick and Leenhouts, Green and Burki, Wideröe. As an example, the Chadwick and Leenhouts generalized model parameters are fitted to a family of experimental survival curves concerningChlorella cells exposed to fractionated and continuous gamma irradiation. The fittings are presented with their confidence limits and are briefly discussed.     

A comparison of the composite and repeat pressurization methods of pressure–volume analysis for shoots of four North American conifer species

Significantly different water relations attributes were derived for temperate conifers measured using the repeat pressurization (RP) and composite (CM) pressure–volume (PV) procedures. In the RP method, single shoots were measured 10–20 times for xylem water potential and mass during air-drying to produce each PV curve. In contrast, for CM PV curves 25–30 shoots were air-dried to relative water contents (R) ranging from 1.0 to 0.5 before being pressurized once. Aggregation of these 25–30 paired values produced single PV curves. Pinus banksiana, P. resinosa and Picea mariana, but not Pinus strobus, had lower full turgor osmotic potential, shallower slope of the linear segment of the PV curves and higher symplast fraction with the CM method. Data points along the linear segment of PV curves were obtained to lower R using the CM method. Reanalysis using similar R ranges eliminated differences between PV methods for Picea mariana but not Pinus banksiana and P. resinosa.     

Phototropism of maize coleoptiles Influences of light gradients

The lateral fluence-rate gradients in unilaterally irradiated maize (Zea mays L.) coleoptiles were calculated on the basis of the proportions of P _fr (far-red-absorbing form of phytochrome) measured spectroscopically in transverse slices of the coleoptiles (top 1 cm). The results showed the occurrence of significant gradients that are wavelength-dependent. The gradient at 449 nm was steeper than those measured at 516, 534 and 551 nm, which were steeper than that measured at 665 nm. The ratios between the sides proximal and distal to the light source were, for example, 1:0.12 (449 nm), 1:0.23 (534 nm), and 1:0.28 (665 nm). Fluence-response curves for coleoptile phototropism (first positive curvature produced by less than 100 s unilateral irradiation) were measured at 449, 516, 534 and 551 nm. Comparison of the threshold fluences indicated that the responsiveness to 551 nm is about 10^4.8 less than that to 449 nm. Increasing wavelengths led to a decrease in maximal curvature, which correlated with the decrease of the fluence-rate ratios between the proximal and distal sides. Phototropic fluence-response curves were also measured using bilateral irradiation (449 nm). In one set of experiments, the fluence ratio was kept constant (either 1:1/2, 1:1/4 or 1:1/16) and the total fluence was varied, and in the other set the fluence applied to one side was kept constant and the fluence ratio was varied. A simple model based on the assumption that only one photoreaction occurs, and that the response is a function of the difference between the proximal and distal sides in the local photoreceptor action was tested. A fluence-response curve for this local photoreceptor action was calculated based on the fluence-rate ratio and the phototropic fluence-response curve measured for 449 nm. This curve was used, in conjunction with the measured fluence-rate ratios, as a basis for calculating phototropic fluence-response curves for other wavelengths and those for 449 nm obtained with bilateral irradiation. The calculated fluence-response curves showed excellent agreement with the experimental data. It is concluded that the threshold for maize coleoptile phototropism reflects the apparent photoconversion cross-section of the blue-light receptor whereas the maximal curvature depends on the steepness of the light gradient across the coleoptile.Abbreviations and symbols I(x) fluence rate at the depth x - P _fr phytochrome (far-red absorbing) - P _r phytochrome (red absorbing) - P _tot total phytochrome (P _r+P _fr) - photoconversion cross-section     

Influence of base-pair changes and cooperativity parameters on the melting curves of short DNAs

Theoretical melting curves were calculated for four DNA restriction fragments, 157–257 base pairs (bp), and a series of hypothetical block DNAs with sequences d(C_2xA_xC_2x). d(C_2xT_xG_2x), 5 ? x ? 40. These DNAs provided a mixture of A·T/G·C sequence distributions with which to investigate the effects of parameters and base-pair changes on the melting of short DNAs. The sensitivity of DNA melting curves to changes in internal loop melting parameters σ and κ was examined. As Expected, theoretical melting curves of short DNAs with a quasirandom base-pair sequence vary little with changes in internal loop parameters. End melting dominates the transition behaviour of these moleucles. This was also observed for the block DNAs up to x = 22. Beyond this length, melting curves are highly sensitive to the internal loop parameters. Sensitivity is also predicted for a 157-bp fragment with a block distribution of A·T and G·C pairs. These results indicate that accurate evaluation of internal loop parameters is possible with short DNAs (100–200 bp) containing a G·C/A·T/G·C block distribution with at least 22 bp in each block. Duplex-to-single-strands dissociation parameters were reevaluated form experimental melting curve data of eight DNA fragments using a least squares fit approach. This analysis confirmed parameter values previously found with a simplified dissociation model. A Priori predictions are made on the effects of base-pair changes on the melting curves of three characterized DNA restriction fragments. Single base-pair changes are predicted to induce small but measurable changes in the melting curves. The characteristics of the altered melting curves depend on the location of the base-pair change.