Description of the delayed microbial growth by an extended logistic equation

Logistic equations are suitable for describing microbial growth. By means of VERHULST'S logistic equation, the adaptation to sigmoid-shaped curves of growth improves with a falling ratio C_xo/C_{x, max} < 0.2, if there is no lag-phase. The known logistic equations do not take into account any lag-phase behaviour, so that noticeable deviations in the model adaptation result in this range. Therefore, an extended logistic equation of rate is proposed by which any occuring lag-time is expressed by a 1st-order lag-term. The corresponding time law allows a very good adaptation of curves of delayed growth behaviour, and changes into VERHULST'S logistic equation for a lag-time t_L = 0. Application is facilitated by instructions for the numerical determination.     

Comparison of parameters estimated from A/Ci and A/Cc curve analysis

The parameters estimated from traditional A/C _i curve analysis are dependent upon some underlying assumptions that substomatal CO₂ concentration (C _i) equals the chloroplast CO₂ concentration (C _c) and the C _i value at which the A/C _i curve switches between Rubisco- and electron transport-limited portions of the curve (C _i-t) is set to a constant. However, the assumptions reduced the accuracy of parameter estimation significantly without taking the influence of C _i-t value and mesophyll conductance (g _m) on parameters into account. Based on the analysis of Larix gmelinii’s A/C _i curves, it showed the C _i-t value varied significantly, ranging from 24 Pa to 72 Pa and averaging 38 Pa. t-test demonstrated there were significant differences in parameters respectively estimated from A/C _i and A/C _c curve analysis (p<0.01). Compared with the maximum ribulose-1,5-bisphosphate carboxylase/oxygenase (Rubisco) carboxylation rate (V_cmax), the maximum electron transport rate (J_max) and J_max/V_cmax estimated from A/C _c curve analysis which considers the effects of g _m limit and simultaneously fits parameters with the whole A/C _c curve, mean V_cmax estimated from A/C _i curve analysis (V_cmax-C _i) was underestimated by 37.49%; mean Jmax estimated from A/C _i curve analysis (J_max-C _i) was overestimated by 17.8% and (J_max-C _i)/(V_cmax-C _i) was overestimated by 24.2%. However, there was a significant linear relationship between V_cmax estimated from A/C _i curve analysis and V_cmax estimated from A/C _c curve analysis, so was it J_max (p<0.05).     

Kinetic study on unzippering of polynucleotides and order-order transition in polypeptides

Solutions are presented for N + 1 sequential and reversible first-order reactions for which the magnitude of the reverse rate constant, k_b, for all steps except the last is identical. Also the magnitude of the forward rate constant, k_f, for all steps except the first and last is identical. The initial and final steps are nucleation reactions; therefore, the initial and final k_f are modified by the factors σ′ and γ respectively. The final k_b is modified by the factor γ σ. The ratio k_b/k_f is defined as s, which has the same meaning as s in the Zimm-Bragg theory. The mathematical model is intended to apply to polymeric molecules of N segments and allows the calculation of the mole fraction of molecules in state i at any time t, C_i(t). A molecule in state i has i unreacted segments and N – i reacted ones. Because the reactions are sequential, all reacted segments are contiguous. Our numerical results show that when σ′ is much less than unity and the forward reaction is favored, the relaxation curve is sigmoidal. If, however, the forward and reverse reactions are equally favored (i.e., s ? 1) the relaxation curve is a straight line. When s and σ′ are near unity, the curve is exponential for a considerably large fraction of the reaction. Further, in the exponential for a considerably large fraction of the reaction. Further, in the exponential phase of the reaction, the relaxation time is proportional to N² for highly cooperative systems (i.e., Nσ ? 1). As found by Pipkin and Gibbs, if N is sufficiently large and s is less than unity (e.g., N ? 50 and s ?0.9) the relaxation curve is largely linear with a slope inversely proportional to N. Applications are given for the unwinding of double-helical poly(A·U) and the order–order transition in poly-L -proline.     

A New Generalized Logistic Sigmoid Growth Equation Compared with the Richards Growth Equation

A new sigmoid growth equation is presented for curve-fitting,analysis and simulation of growth curves. Like the logisticgrowth equation, it increases monotonically, with both upperand lower asymptotes. Like the Richards growth equation, itcan have its maximum slope at any value between its minimumand maximum. The new sigmoid equation is unique because it alwaystends towards exponential growth at small sizes or low densities,unlike the Richards equation, which only has this characteristicin part of its range. The new sigmoid equation is thereforeuniquely suitable for circumstances in which growth at smallsizes or low densities is expected to be approximately exponential,and the maximum slope of the growth curve can be at any value.Eleven widely different sigmoid curves were constructed withan exponential form at low values, using an independent algorithm.Sets of 100 variations of sequences of 20 points along eachcurve were created by adding random errors. In general, thenew sigmoid equation fitted the sequences of points as closelyas the original curves that they were generated from. The newsigmoid equation always gave closer fits and more accurate estimatesof the characteristics of the 11 original sigmoid curves thanthe Richards equation. The Richards equation could not estimatethe maximum intrinsic rate of increase (relative growth rate)of several of the curves. Both equations tended to estimatethat points of inflexion were closer to half the maximum sizethan was actually the case; the Richards equation underestimatedasymmetry by more than the new sigmoid equation. When the twoequations were compared by fitting to the example dataset thatwas used in the original presentation of the Richards growthequation, both equations gave good fits. The Richards equationis sometimes suitable for growth processes that may or may notbe close to exponential during initial growth. The new sigmoidis more suitable when initial growth is believed to be generallyclose to exponential, when estimates of maximum relative growthrate are required, or for generic growth simulations.Copyright1999 Annals of Botany Company Asymptote,Cucumis melo,curve-fitting, exponential growth, intrinsic rate of increase, logistic equation, maximum growth rate, model, non-linear least-squares regression, numerical algorithm, point of inflexion, relative growth rate, Richards growth equation, sigmoid growth curve.     

The CO2-concentrating mechanism is absent in the green alga Coccomyxa: a comparative study of photosynthetic CO2 and light responses of Coccomyxa, Chlamydomonas reinhardtii and barley protoplasts

Photosynthesis was characterized for the unicellular green alga Coccomyxa sp., grown at low inorganic carbon (C_i) concentrations, and compared with Chlamydomonas reinhardtii, which had been grown so that the CO₂ concentrating mechanism (CCM) was expressed, and with protoplasts isolated from the C₃ plant barley (Hordeum vulgare). Chlamydomonas had a significantly higher C_i-use efficiency of photosynthesis, with an initial slope of the C_i-response curve of 0.7 mol(gChl)⁻¹ h⁻¹ mmol C_im⁻³)⁻¹, as compared to 0.3 and 0.23 mol(gChl)⁻¹ h⁻¹ (mmol C_im⁻³)⁻¹ for Coccomyxa and barley, respectively. The affinity for C_i was also higher in Chlamydomonas, as the half maximum rate of photosynthesis [K_0.5 (C_i)] was reached at 0.18 mol m⁻³, as compared to 0.30 and 0.45 mol m⁻³ for Coccomyxa and barley, respectively. Ethoxyzolamide (EZ), an inhibitor of the enzyme carbonic anhydrase (CA) and the CCM, caused a 17-fold decrease in the initial slope of the photosynthetic Cj-response curve in Chlamydomonas, but only a 1.5- to two-fold decrease in Coccomyxa and barley. The photosynthetic light-response curve showed further similarities between barley and Coccomyxa. The rate of bending of the curve, described by the convexity parameter, was 0.99 (sharp bending) and 0.81–0.83 (gradual bending) for cells grown under low and high light, respectively. In contrast, the maximum convexity of Chlamydomonas was 0.85. The intrinsically lower convexity of Chlamydomonas is suggested to result from the diversion of electron transport from carbon fixation to the CCM. Taken together, these results suggest that Coccomyxa does not possess a CCM and due to this apparent lack of a CCM, we propose that Coccomyxa is a better cell model system for studying C₃ plant photosynthesis than many algae currently used.     

Analysis of starch amylolysis using plots for first-order kinetics

Investigators often study product release from starches during prolonged incubations with α-amylase in vitro. The reaction time courses usually fit to a linear form of a first order rate equation, i.e., ln[(C_∞ − C_t)/C_∞] = −kt. This equation calls for an accurate estimate of C_∞, i.e., the concentration of product at the end of the reaction. Estimates of C_∞ from digestibility curves can be unreliable. The Guggenheim method does not require prior knowledge of C_∞ but seems not to have been applied to starch hydrolysis data. An alternative method is also available in which the logarithm of the slope (LOS) of a digestibility curve at various time points is plotted against time. This allows estimations of both k and C_∞ and can also reveal whether changes occur in digestion rate from rapid to slow as digestion proceeds. We describe the Guggenheim and LOS methods and provide examples of their application to starch digestibility data.     

Effect of jasmonic acid on the stomatal and nonstomatal limitation of leaf photosynthesis in barley leaves

The effect of long-term (7 days) and shortterm (up to 2 h) treatment of barley plants with jasmonic acid (JA) on the components contributing to stomatal and nonstomatal limitation of photosynthesis was studied. Net CO₂ assimilation rate (A) responses to intercellular CO₂ concentration (C _i ), i.e., A/C _i curves, were used to assess the photosynthetic ability. Long-term treatment of barley plants with JA led to a noticeable decrease in both the initial slope of the A/C _i curves and the maximum A at saturating C _i . The proportion of stomatal and nonstomatal factors in limitation of photosynthesis depended on the applied JA concentration. Short-term treatment with JA affected neither the stomatal conductivity for CO₂ nor the rate of photosynthetic CO₂ assimilation. We suggest that JA may affect photosynthesis indirectly, either as a stress-modulating substance, or through the alterations in gene expression.     

The fitting of the logistic equation to the rate of increase of population density

Though there are many problems on the usefulness of the logistic curve, it may be necessary to examine before discussing these problems whether or not the actual data fit to the theoretical values. It has been clarified in this paper that the relation between the population density and its rate of increase per individual described by the differential equation (1) is represented by a straight line on a finite difference diagram on which N_i+1−N_i/N_i values are plotted against N_i+1. Utilizing this linear relation we may examine the fittness of the logistic curve to the actual data and when it is fitted we may estimate the parameters of the logistic equation by (5) and (6). The result of the application of this method to the experimental populations of azuki bean weevil indicates that the relation between parent and progeny densities fits well to the logistic type as has been proved byFujita andUtida (1953) who utilized the linear reltion between 1/R+2σ and parent density where R is the apparent rate of reproduction and σ is a constant dependent primarily upon the length of adult life (0≦σ≦1).     

A statistical model of the reproduction of aphids

1. The logistic function has been generally used to describe the reproductive process of a “population” of animal. However, this model can not give us any information about the reproductive process of “individuals” in the population. In this study a statistical model on the basis of the reproduction of individuals of barley aphid is presented to find the proportion of the mature individuals, the heterogeneity in reproductive ability of the aphids, etc.
2. The model is constructed as follows:
3. The probability that j insects are found on a plant at time t₀ is represented as Q(j).
4. The probability that h individuals of j have reproductive ability, say, mature individuals, in the period t₀ to t₁ is represented as B(h/j)=_jC_hw^h(1−w)^j−h, where w is the proportion of mature individuals.
5. In a population with a homogeneous reproductive ability, the probability that each parent lays i offspring in the period t₀ to t₁ is represented as P(i/m)=e^−mmⁱ/i!, where m is mean. And, in a population, m changes according to the gamma distribution. Hence the probability that a parent lays i offspring between t₀ and t₁ is represented as , where p and k are parameters of negative binomial distribution. The probability that h parents on a plant lays s offspring is represented as .
6. From the assumptions mentioned above, the probability that s offspring are to be found at time t₁ on a plant with the original j individuals at time t₀ is represented by
7. The experimental populations were demonstrated to fit well to the model.

     

A model of the plant-to-plant movement of aphids I. Description of the model

The plant-to-plant movement of insects in one of the factors determining the distribution of individuals in insect populations. In this report the movement of barley aphids was analyzed by a statistical model. The model is represented as the convolution of three probability functions:

1. The probability that s individuals are found on a plant at time t₀:Q(s);
2. The probability that i individuals leave the plant and remain on the ground from time t₀ to t₁:_sC_ipⁱq^s−i and p+q=1, where p and q are the proportions of individuals which do not leave a plant and which leave it once or more, respectively;
3. The probability that j individuals climb a plant between time t₀ to t₁ and stay there at time t₁:e^−λλ^j/λ^!, where λ is the mean of the individuals.

The probability that l individuals are located on a plant at time t₁ is represented by the following equation It was shown by simple experiments that the experimental populations were well fitted to the model.     

Thermochemical Characterization of the Growth Metabolism of the 2,4-Dichlorophenol-Degrading Strain Pseudomonas GT241-1 by Microcalorimetry

By using an LKB-2277 Bioactivity Monitor and the ampoule method, the heat output of the growth metabolism of a 2,4-dichlorophenol-degrading bacterial strain, Pseudomonas strain GT241-1, has been determined at 30°C. From the thermogenic curves, it can be established that the thermokinetic equation of their growth metabolism is P _t = P _{t = 0} exp(k _m t), dP/dt = k _m P ¹, with the order of growth metabolism n = 1. The experimental results indicate that the relationship between the metabolic power (P) and the cell concentration (C) and the relationship between the metabolic power of each cell (P ₀) and the cell concentration can be characterized by the following thermal equations, respectively: C = a + kP and lnC = a+kP ₀ or d dC/dP ₀ = KC ¹. The order of the P ₀ –C equation n is also 1. These results are very significant for environmental sciences, biology, and thermochemistry.     

Computer-determined rate constants of hemolysis induced by Prymnesium parvum toxin

Rates of hemolysis of rabbit erythrocyte suspensions induced by P. parvum (prymnesin) have been measured colorimetrically at 25.5°C and pH 5.5. The data have been treated previously as consecutive first-order rate processes associated with the prolytic and lytic periods from which two specific rate constants have been obtained, k′ and kψ, respectively. These constants have been related to those obtained by a computer-generated fit of the rate data (absorbance A_t, as a function of time t) with the rate equation $\text{Y =}\text{D}\text{[1 +}\text{exp}\text{((X ? B)}\text{C)}\text{]}\text{+ E}$. Here Y equals A_t, X = time, t; D is equal to a spread factor, A_i ? A_∞; C is the slope of the curve at the inflection point; B is the midpoint time value, i.e., the time at which $\text{A}{}_{\mathrm{t}}\text{=}\text{D}\text{2}$; E is termed the off-set constant and is equal to A_∞. Of these constants, B is directly related to the length of the prolytic period, and C^?1 is directly related to the specific first-order rate constant for hemolysis, k_ψ.     

Kinetic Analysis of the Noncompetitive Inhibition of the Lignin-Peroxidase-catalyzed Reaction by Oxalic Acid

Oxalic acid was found to inhibit noncompetitively the Cα-Cβ bond cleavage of veratrylglycerol catalyzed by a lignin peroxidase (LiP) isozyme of the white-rot fungus P. chrysosporium. With greater amounts of oxalic acid in the LiP system, the substrate was not converted to veratraldehyde but was almost all recovered. Oxalic acid was shown to be decomposed to CO₂ during the enzymatic reaction. The results clearly indicate that oxalic acid reduced the cation radical intermediate formed in the reaction back to the substrate to block the production of veratraldehyde. A novel equation has been derived to explain the mechanism for this unique non-competitive inhibition that is different from the classical noncompetitive one. The inhibition constant K_i obtained here, which is different from the classical inhibition constant K_i, is defined as the ratio of the rate constant (k_p) for product formation to the rate constant (k_i) for the reduction of the cation radical to the substrate.     

Improved method for measuring the apparent CO2 photocompensation point resolves the impact of multiple internal conductances to CO2 to net gas exchange

There is a growing interest in accurate and comparable measurements of the CO₂ photocompensation point (Γ*), a vital parameter to model leaf photosynthesis. The Γ* is measured as the common intersection of several CO₂ response curves, but this method may incorrectly estimate Γ* by using linear fits to extrapolate curvilinear responses and single conductances to convert intercellular photocompensation points (C_i*) to chloroplastic Γ*. To determine the magnitude and minimize the impact of these artefacts on Γ* determinations, we used a combination of meta‐analysis, modelling and original measurements to develop a framework to accurately determine C_i*. Our modelling indicated that the impact of using linear fits could be minimized based on the measurement CO₂ range. We also propose a novel method of analysing common intersection measurements using slope–intercept regression. Our modelling indicated that slope–intercept regression is a robust analytical tool that can help determine if a measurement is biased because of multiple internal conductances to CO₂. Application of slope–intercept regression to Nicotiana tabacum and Glycine max revealed that multiple conductances likely have little impact to C_i* measurements in these species. These findings present a robust and easy to apply protocol to help resolve key questions concerning CO₂ conductance through leaves.     

Inhibition of extracellular lipase from Streptomyces rimosus with 3,4-dichloroisocoumarin

Kinetic characterization of lipase inhibition was performed by activity measurement and mass spectrometry (MS), for the first time with serine-protease inhibitor 3,4-dichloroisocoumarin (DCI). Inhibition of Streptomyces rimosus extracellular lipase (SrLip), a member of the SGNH superfamily, by means of DCI follows the mechanism of two-step irreversible inhibition. The dissociation constant of the noncovalent E?I complex and first-order rate constant for inactivation were determined by incubation (K_i* = 26.6?±?2.8 µM, k₂ = 12.2?±?0.6 min–1) or progress curve (K_i* = 6.5?±?1.5 µM, k₂ = 0.11?±?0.01 min–1) method. Half-times of reactivation for lipase inhibited with 10-fold molar excess of DCI were determined by activity measurement (t_1/2 = 11.3?±?0.2?h), matrix-assisted laser desorption/ionization (MALDI, t_1/2 = 13.5?±?0.4?h), and electro-spray ionization (ESI, t_1/2 = 12.2?±?0.5?h) MS. The active SrLip concentration was determined by incubating the enzyme with near equimolar concentrations of DCI, followed by activity and MS measurement.     

Disintegration of yeast Saccharomyces cerevisiae in the vertical perl mill with a bell-shaped impeller

Investigation of disintegration of yeast Saccharomyces cerevisiae in the laboratory batch perl mill with a bell-shaped impeller was carried out. The number of non-damaged cells, changing in time was determined using hemocytometer (Thom's chamber).To describe kinetics of the disintegration process the differential equation was applied: where N _p the number of non-damaged cells in the sample, [number of cells/ml] t time, [s] m,k constants.The effect of three operating parameters: rotation frequency of the impeller shaft n, filling of the mill with disintegrating elements (ballotini) S _k and the initial concentration of yeast cells in the suspension C ₀ on the process of disintegration was analyzed.For S _k =0.5, m=1 and dependence of constant k on the rotation frequency of the impeller and suspension concentration were obtained. For S _k =0.6 and 0.7 the values of m were higher than 1. The effect of rotation frequency of the impeller and filling of the mill, with ballotini on constant k and exponent m was determined.List of Symbols a, b constants - a ₁, b ₁, c ₁, d ₁ constants - C ₀ initial concentration of suspension g/ml - C concentration of cell suspension g/ml - k constant disintegration rate 1/s; N ₀ ^1-m /s - m variable in the equation - N ₀ initial number of cells no. of cells/ml - N _p number of non-damaged cells no. of cells/ml - r process rate g/ml·s - X(t) disintegration degree % - , variables in the equation - z variable in the equation - S _k degree of filling the mill with disintegrating elements     

On the need to incorporate sensitivity to CO2 transfer conductance into the Farquhar–von Caemmerer–Berry leaf photosynthesis model

Virtually all current estimates of the maximum carboxylation rate (V_cmax) of ribulose‐1,5‐bisphosphate carboxylase/oxygenase (Rubisco) and the maximum electron transport rate (J_max) for C₃ species implicitly assume an infinite CO₂ transfer conductance (g_i). And yet, most measurements in perennial plant species or in ageing or stressed leaves show that g_i imposes a significant limitation on photosynthesis. Herein, we demonstrate that many current parameterizations of the photosynthesis model of Farquhar, von Caemmerer & Berry (Planta 149, 78–90, 1980 ) based on the leaf intercellular CO₂ concentration (C_i) are incorrect for leaves where g_i limits photosynthesis. We show how conventional A–C_i curve (net CO₂ assimilation rate of a leaf –A_n– as a function of C_i) fitting methods which rely on a rectangular hyperbola model under the assumption of infinite g_i can significantly underestimate V_cmax for such leaves. Alternative parameterizations of the conventional method based on a single, apparent Michaelis–Menten constant for CO₂ evaluated at C_i[K_m(CO₂)_i] used for all C₃ plants are also not acceptable since the relationship between V_cmax and g_i is not conserved among species. We present an alternative A–C_i curve fitting method that accounts for g_i through a non‐rectangular hyperbola version of the model of Farquhar et al. (1980 ). Simulated and real examples are used to demonstrate how this new approach eliminates the errors of the conventional A–C_i curve fitting method and provides V_cmax estimates that are virtually insensitive to g_i. Finally, we show how the new A–C_i curve fitting method can be used to estimate the value of the kinetic constants of Rubisco in vivo is presented     

Multiple source pools for Galápagos plant species richness: a critical analysis of the line of sight connectivity index

An index (C_i*E) combining the number of line‐of‐sight islands (C_i) within a radius i and target island elevation (E) has been proposed as an improved predictive model of plant species richness (S_t) in the Galápagos Archipelago. We examined this index critically and found that several major flaws preclude it from being a useful predictive tool for the archipelago. Although the number of collecting trips to an island was reported over 20 years ago to have substantial predictive value for reported plant species richness in the Galápagos Islands, this relationship was ignored in multiple regression analyses of the index. When we included the number of collecting trips in different multiple regression analyses of the index, C_i*E had less predictive power than collecting trips or ceased to be significant at all. Additionally, the strong significant relationship between elevation and area in the Galápagos Archipelago results in area having a major confounding influence on the C_i*E index. When elevation is removed from the C_i*E index, the predictive power of C_i is far less than area alone. Finally, the data used to construct and correlate the C_i*E index with (S_t) were based only on a subset of the islands and species lists that were incomplete or out of date. Species richness on islands can be related to the interaction of different factors, so development and testing of indices like C_i*E is not inappropriate. However, it is important to examine the interrelationships among the components of these indices thoroughly, and not ignore the effect of factors already known to have high predictive power. We propose several ways in which more meaningful indices of source pool(s) capacity can be constructed.     

Analysis of the ionization constants and heats of ionization of reduced and oxidized horse heart cytochrome c

Simultaneous curve fitting for the ionization parameters of oxidized and reduced horse heart cytochrome c in 0.15M KCl and 20°C yields values for the ionization constants (as pK′) and the heats of ionization (ΔH_i) which can reconstruct either the potentiometric or thermal titration curves. Reduced cytochrome c requires 8 sets of groups, whereas oxidized cytochrome c requires 10 sets of groups. The additional groups in the oxidized preparation appear to involve the ferriheme (pK′, 9.25; ΔH_i, 13.7 kcal/mol) and a tyrosine (pK′ ? 10.24) that is not present in the reduced form. The potentiometric and thermal difference curves (reduced – oxidized) involve the appearance of 17 kcal/mol centered at pH 9.7 and 5.8 kcal/mol centered at pH 4.9. The carboxyl groups in both species appear to be normal for the hydrogen-bonded form. Only one histidine has normal ionization properties (pK′, 6.7; ΔH_i, 7.5 kcal/mol), as do 17 of the lysine residues (pK′, 10.8; ΔH_i, 11.5 kcal/mol).