Age structure in predator-prey systems: Intraspecific carnivore interaction,passive diffusion,and the paradox of enrichment

An existing arthropod predator-prey model incorporating age structure in the carnivore through the use of the von Foerster equation is extended to include the effects of intraspecific carnivore interaction and passive diffusion or migration. A linear stability analysis of the community equilibrium point of that differential-integral equation system is performed and the resulting secular equation analyzed by the method of D-partitions. These stability results are then compared to those obtained by employing an analogous differential equation model without age structure, in particular as they relate to the so-called paradox of enrichment. In the absence of passive diffusion, it is shown that, unlike for a differential equation model, the paradox of enrichment can occur even with a carnivore which exhibits intraspecific competition. This destabilizing effect of age structure is seen to occur most dramatically when interspecific interactions are large, while the effect of passive diffusion is to offset that tendency and restabilize the system. These predictions are in accordance with relevant experimental evidence involving mites.     

Equations and calculations for fermentations of butyric acid bacteria. Reprinted from Biotechnology and Bioengineering, Vol. XXVI, Pp 174-187 (1984)

A stoichiometric equation has been derived which describes the interrelations among the various products and biomass in fermentations of butyric acid bacteria. The derivation of the equation is based on an assumed ATP yield, two biological regularities, and the biochemistry of product formation of the fermentations. The equation obeys the constraints imposed on growth and product formation by thermodynamics and the biochemical topology. The validity of the equation is tested using a variety of fermentation data from the literature. The uses, improvements, limitations, and extensions of the equation are also discussed in detail. For example, the fermentation equation is used to calculate the maximal possible yields of the main fermentation products.     

Grafen,the Price equations,fitness maximization,optimisation and the fundamental theorem of natural selection

This paper is a commentary on the focal article by Grafen and on earlier papers of his on which many of the results of this focal paper depend. Thus it is in effect a commentary on the “formal Darwinian project”, the focus of this sequence of papers. Several problems with this sequence are raised and discussed. The first of these concerns fitness maximization. It is often claimed in these papers that natural selection leads to a maximization of fitness and that this view is claimed in Fisher’s “fundamental theorem of natural selection”. These claims are refuted, and various incorrect statements about the meaning and interpretation of the fundamental theorem of natural selection, in this sequence and in other papers by other authors, are discussed. Next, much of the work in this sequence rests on the first Price equation. In the deterministic (infinite population) case this equation is no more than the standard classical equation relating to changes in gene frequencies. In the stochastic case the equation gives the change in gene frequencies as the sum of two terms (the second of which vanishes in the deterministic case). These two terms are of essentially equal importance in the situation considered in the focal article, yet one of Grafen’s results ignores the second term in the stochastic analysis. This is associated with a wavering between deterministic and stochastic analyses and the use of the Price fitness concept and the classical fitness concept. These comments cast doubts on Grafen’s optimization theory.     

Extended monod kinetics for substrate, product, and cell inhibition

A generalized form of Monod kinetics is proposed to account for all kinds of product, cell, and substrate inhibition. This model assumes that there exists a critical inhibitor concentration above which cells cannot grow, and that the constants of the Monod equation are functions of this limiting inhibitor concentration. Methods for evaluating the constants of this rate form are presented. Finally the proposed kinetic form is compared with the available data in the literature, which unfortunately is very sparse. In all cases, this equation form fitted the data very well.     

A generic rate equation for catalysed,template-directed polymerisation

Biosynthetic networks link to growth and reproduction processes through template-directed synthesis of macromolecules such as polynucleotides and polypeptides. No rate equation exists that captures this link in a way that it can effectively be incorporated into a single computational model of the overall process. This paper describes the derivation of such a generic steady-state rate equation for catalysed, template-directed polymerisation reactions with varying monomer stoichiometry and varying chain length. The derivation is based on a classical Michaelis–Menten mechanism with template binding and an arbitrary number of chain elongation steps that produce a polymer composed of an arbitrary number of monomer types. The rate equation only requires the identity of the first dimer in the polymer sequence; for the remainder only the monomer composition needs be known. Further simplification of a term in the denominator yielded an equation requiring no positional information at all, only the monomer composition of the polymer; this equation still gave an excellent estimate of the reaction rate provided that either the monomer concentrations are at least half-saturating, or the polymer is very long.     

Studies in the mathematical biophysics of discrimination and conditioning II: Special case: Errors,trials, and number of possible responses

A special case of a problem discussed in a previous paper is treated in greater detail. An equation in the three variables, errors, trials and number of possible choices, is developed and compared with the results of an experiment performed under conditions closely approximating those required for the development of the equation. The agreement is excellent.     

黑襟毛瓢虫对棉蚜的数值反应

本文研究了不同龄期及不同发育期的黑襟毛瓢虫在不同食蚜量(生物量)下的发育速率,增长速率,存活率及生殖率。主要结果如下: (1)讨论了下模型(Beddinton et al.,1976)的隐含假说: 1/d=a(KaNT/1 aT _(?)N-B) (1) 其中最主要一点就是该模型没有考虑到在(食物摄入量)常数B邻域内发育速率的非连续性。为了描述此性质,我们提出了以下通式: 式中:1/d是捕食者发育速率,X是捕食者摄取的生物量,N为猎物密度。F(X(N))为一可以是线性或其他形状的函数。笔者认为发育速率在B领域的不连续性是数值反应的一个普遍规律。(2)黑襟毛瓢虫幼虫取食量与增长率的关系结果表明,三、四龄幼虫在取食量较低时能以低于上一龄期初的体重(负增长率)进入下一龄期或虫态。(3)二、三龄幼虫有很强的耐饥性,但一、四龄幼虫对饥饿较敏感(图5)。第i—1龄幼虫的取食量能影响第i龄时的存活率及发育速率(表2)。(4)幼虫及成虫的取食率对生殖均有重要影响,但取食量对卵重及其卵的孵化率影响不显著。     

Comparing the regulatory behaviour of two cooperative, reversible enzyme mechanisms

It is shown that both the reversible Hill equation and a generalised, reversible Monod-Wyman-Changeux equation can give analogous regulatory behaviour when embedded in a model metabolic pathway.     

The use of a metabolically structured model in the study of growth, nitrification, and denitrification by Thiosphaera pantotropha

Thiosphaera pantotropha is capable of aerobic heterotrophic nitrification and both aerobic and anaerobic denitrification. These phenomena have been studied in acetate-limited aerobic and anaerobic continuous cultures supplied with ammonia and nitrate. The internal reaction rates were defined, based on biochemical knowledge. The observable external conversion rates are related through a linear equation on the basis of the specified internal reaction rates. The linear equation is a Pirt relation extended for microbial systems with multiple electron donors (acetate and ammonia) and electron acceptors (oxygen and nitrate). The coefficients in this equation were estimated from the continuous culture measurements, and are composed of parameters involved in ATP production and consumption by the microorganism. It is shown that with realistic values for these parameters, the metabolically structured model describes the aerobic as well as the anaerobic experiments.     

A classic learning opportunity from Fenn, Rahn, and Otis (1946): the alveolar gas equation

The alveolar gas equation, the focus of a classic paper by Fenn, Rahn, and Otis, provides a framework for understanding the mechanisms involved in pulmonary gas exchange as well as the limits of human performance. The classic 1946 paper by Fehn, Rahn, and Otis gives your students an opportunity to learn about the alveolar gas equation from the physiologists who pioneered it and demonstrates that mathematics and data graphics are fundamental tools with which to learn respiratory physiology. In this essay, I outline avenues of discovery by which your students can explore the alveolar gas equation. Meaningful learning stems from inspiration: to learn, you must be inspired to learn. If anyone can inspire learning in respiratory physiology, it is Wallace Fenn, Hermann Rahn, and Arthur Otis.     

Simultaneous estimation ofV max,Km, and the rate of endogenous substrate production (R) from substrate depletion data

The nonlinear and 3 linearized forms of the integrated Michaelis-Menten equation were evaluated for their ability to provide reliable estimates of uptake kinetic parameters, when the initial substrate concentration (S₀) is not error-free. Of the 3 linearized forms, the one where t/(S₀–S) is regressed against ln(S₀/S)/(S₀–S) gave estimates ofV _max and K_m closest to the true population means of these parameters. Further, this linearization was the least sensitive of the 3 to errors (±1%) in S₀. Our results illustrate the danger of relying on r² values for choosing among the 3 linearized forms of the integrated Michaelis-Menten equation. Nonlinear regression analysis of progress curve data, when S₀ is not free of error, was superior to even the best of the 3 linearized forms. The integrated Michaelis-Menten equation should not be used to estimateV _max and K_m when substrate production occurs concomitant with consumption of added substrate. We propose the use of a new equation for estimation of these parameters along with a parameter describing endogenous substrate production (R) for kinetic studies done with samples from natural habitats, in which the substrate of interest is an intermediate. The application of this new equation was illustrated for both simulated data and previously obtained H₂ depletion data. The only means by whichV _max, K_m, and R may be evaluated from progress curve data using this new equation is via nonlinear regression, since a linearized form of this equation could not be derived. Mathematical components of computer programs written for fitting data to either of the above nonlinear models using nonlinear least squares analysis are presented.     

LDL-C calculated by Friedewald,Martin-Hopkins,or NIH equation 2 versus beta-quantification: pooled alirocumab trials

Accurate assessment of LDL-C levels is important, as they are often used for treatment recommendations. For many years, plasma LDL-C levels were calculated using the Friedewald equation, but there are limitations to this method compared with direct measurement via beta-quantification (BQ). Here, we assessed differences between the Friedewald, Martin-Hopkins, and NIH equation 2 methods of calculating LDL-C and the “gold standard” BQ method using pooled phase 3 data with alirocumab. All randomized patients were included irrespective of the treatment arm (n = 6,122). We compared pairs of LDL-C values (n = 17,077) determined by each equation and BQ. We found that BQ-derived LDL-C values ranged from 1 to 397 mg/dl (mean 90.68 mg/dl). There were strong correlations between Friedewald-calculated, Martin-Hopkins–calculated, and NIH equation 2–calculated LDL-C with BQ-determined LDL-C values (Pearson's correlation coefficient = 0.985, 0.981, and 0.985, respectively). Importantly, for BQ-derived LDL-C values ≥70 mg/dl, only 3.2%, 1.4%, and 1.8% of Friedewald-calculated, Martin-Hopkins–calculated, and NIH equation 2–calculated values were <70 mg/dl, respectively. When triglyceride (TG) levels were <150 mg/dl, differences between calculated and BQ-derived LDL-C values were minimal, regardless of the LDL-C level (<40, <55, or <70 mg/dl). However, when TG levels were >150 mg/dl, NIH equation 2 provided greater accuracy than Friedewald or Martin-Hopkins. When TGs were >250 mg/dl, inaccuracies were seen with all three methods, although NIH equation 2 remained the most accurate. In conclusion, LDL-C calculated by any of the three methods can guide treatment decisions for most patients, including those treated with proprotein convertase subtilisin/kexin type 9 inhibitors.     

Use of a serum-based glomerular filtration rate prediction equation to assess renal function by age, sex, fasting, and health status in bottlenose dolphins (Tursiops truncatus)

Glomerular filtration rate (GFR) is a direct measurement of renal function. Although clearance tests using 24‐h urine collection or blood sample series are gold standards for measuring GFR, serum‐based prediction of GFR based upon the Modification of Diet in Renal Disease (MDRD) Study equation is acceptable for routine use in human adults. The purpose of our study was to assess the ability for a modified MDRD Study equation to predict expected changes in GFR in bottlenose dolphins (Tursiops truncatus) using a healthy dolphin population represented by 1,103 routine serum samples collected from 50 dolphins of all age groups, years 1998–2005. Predicted GFR was also calculated from serum collected from a 32‐yr‐old male dolphin with end‐stage renal disease. The dolphin‐adjusted MDRD equation predicted GFR changes in our population that paralleled what has previously been reported in other mammals, including decreasing predicted GFR with age (P < 0.01), higher predicted GFR in dolphins that had recently eaten (P < 0.01), and rapidly decreasing predicted GFR in the animal with end‐stage renal disease. We conclude that a serum‐based GFR prediction equation may be a feasible means of detecting and tracking renal function in bottlenose dolphins.     

The δ-function counterion condensate around a line charge derived from the Fuoss,Katchalsky, and Lifson polyelectrolyte theory

It has recently been proven that the counterion condensate around an isolated line charge in an electrolyte, as characterized by nonlinear Poisson-Boltzmann theory, is an encapsulating δ-function. Here the identical result is shown to hold in the framework of the polyelectrolyte theory of Fuoss, Katchalsky, and Lifson. The proof fully exploits analytic solutions to the differential equation which are not available for the nonlinear, cylindrical Poisson-Boltzmann equation.     

Kinetics of protein-modification reactions. Determination of the fractional concentration of enzyme protein groups, or group reactivities, essential for catalytic function.

A mathematical treatment of modification-induced enzyme protein inactivation is presented, and it is shown that, at initial reaction conditions, the ratio of the first derivative of the equation describing enzyme activity loss to the first derivative of the equation describing protein groups modification is equal to the fractional concentration of enzyme protein reactive groups, or group reactivities, essential for catalytic function.     

The oxygen equilibrium of the hemoglobin of the eel,Anguilla rostrata

Kawamoto had reported that eel hemoglobin has a hyperbolic oxygen equilibrium function, with n in the Hill equation equal to 1. On the basis of Kawamoto's data and with new measurements, it is shown that the equilibrium function is in fact S-shaped, as in most other vertebrates, and n in Hill's equation equals 1.8.     

Cellular calcium signals: nature, registration, and quantitative estimation

Calcium ions play a central role in the regulation of cellular activity. Calcium influx across the plasma membrane occurs through ion channels (voltage- and receptor-operated channels). Two intracellular channels responsible for releasing Ca2+ from the internal stores are ryanodine and IP3 receptors. Two mechanisms for Ca2+ extrusion have been identified in the sarcolemma (Ca2+ pump and Na+/Ca2+ exchanger) and one in the sarcoplasmic membrane (Ca2+ pump). Hierarchical organization of intracellular calcium signalling is presented. It is considered of opening of the single channels or of groups channels to give quarks and sparks. The methods for the determination of the intracellular Ca2+ concentration are discussed. The equation connecting [Ca2+]i with double wavelengths parameter R was obtained proceeding from three fluorescent forms of indo-1 (L, LM and LP). Using this equation permits improving calculation of [Ca2+]i.     

FILTRATION,DIFFUSION, AND MOLECULAR SIEVING THROUGH POROUS CELLULOSE MEMBRANES

In Vol. 38, No. 2, November 20, 1954, page 226, in Equation 2, there is an error in the sign of the last term in the denominator of the logarithm. The equation should read as follows:— See PDF for Equation