THE KINETICS OF SENESCENCE

The course of decline of vitality with age due to the process of senescence, when not complicated by the process of growth, follows a simple exponential law; that is the degree of vitality or of senescence (defining vitality as the reciprocal of senescence) at any moment is, regardless of age, a constant percentage of the degree of vitality or senescence of the preceding moment. This exponential law is the same as the law of monomolecular change in chemistry. During the actively growing period of life the index of vitality rises, due to the process of growth and the course of vitality in the case when the growing period is included in the vitality curve, follows a rising and falling course. This rising and falling course may often be represented by an equation containing two exponential terms which is practically the equation used to represent the course of accumulation and disappearance of a substance as the result of two simultaneous consecutive monomolecular chemical reactions.     

THE REGENERATION OF VISUAL PURPLE IN THE LIVING ANIMAL

1. The accumulation of visual purple in the retina after bleaching by light has been studied in the intact eye of the frog. The data show that duration and intensity of light adaptation, which influence the course of human dark adaptation as measured in terms of visual threshold, have a similar influence on the course of visual purple regeneration. 2. At 25°C. frogs which have been light adapted to 1700 millilamberts and then placed in the dark, show an increase in visual purple concentration which begins immediately and continues for 70 minutes until a maximum concentration is attained. The increase, although beginning at once, is slow at first, then proceeds rapidly, and finally slows up towards the end. Frogs which have been adapted to 9500 millilamberts show essentially the same phenomenon except that the initial slow period is strongly delayed so that almost no visual purple is formed in the first 10 minutes. 3. At 15°C. the initial delay in visual purple regeneration occurs following light adaptation to both 1700 and 9500 millilamberts. The delay is about 10 minutes and is slightly longer following the higher light adaptation. 4. The entire course of visual purple accumulation in the dark takes longer at the lower temperature than at the higher. The temperature coefficient for 10°C. is about 1.8. 5. In contrast to the behavior of the isolated retina which has small amounts of vitamin A and large amounts of retinene immediately after exposure to light, the intact eye has large amounts of vitamin A and little retinene after exposure to light for 10 minutes. In the intact eye during dark adaptation, the amount of vitamin A decreases markedly while retinene decreases only slightly in amount. If retinene is formed in the intact eye, the change from retinene to vitamin A must therefore occur rapidly in contrast to the slow change in the isolated retina. 6. The course of visual purple regeneration may be described by the equation for a first order autocatalyzed reaction. This supposes that the regeneration of visual purple is catalyzed by visual purple itself and accounts for the sigmoid shape of the data.     

THE RATE OF GROWTH OF THE DAIRY COW : V. EXTRAUTERINE GROWTH IN LINEAR DIMENSIONS.

Barring fluctuations due to the cyclic phenomena, the extrauterine course of growth in linear dimensions and in weight of the dairy cow follows an exponential law having the same form as the law representing the course of monomolecular change in chemistry. This suggests the interpretation that the general course of growth is limited by a monomolecular chemical process, and that the cyclic phenomena are due to subsidiary processes in the fundamentally exponential course of growth. The fact that growth follows or tends to follow an exponential course may be stated more simply as follows: if the unit of time is taken sufficiently large so that fluctuations due to the cyclic phenomena are balanced or eliminated, then the amount of growth made during the given unit of time at any age tends to be a constant percentage of the growth made during the preceding unit of time. Thus, the growth in height at withers made during any year is about 34 per cent of the growth made during the preceding year. Similarly the growth in weight made during any year is about 56 per cent of the growth in weight made during the preceding year. This is in accordance with expectations if it is assumed that each animal begins life with a definite endowment of limiting substance necessary for the process of growth, and that this endowment is used up at a constant rate (or percentage) of itself.     

ON THE NATURE OF THE EQUATION FOR GROWTH PROCESSES

An analysis of the growth curves of a cladoceran for one adult instar at each of two temperatures is made by comparing the apparent gains or losses in time when the animals are transferred from one of these temperatures to the other during the course of the developmental period. Since the curves for the two temperatures when brought together at their end-point do not coincide, the equation used to describe growth must have at least two velocity constants unequally affected by changes in temperature.     

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 RATE OF GROWTH OF THE DAIRY COW : IV. GROWTH AND SENESCENCE AS MEASURED BY THE RISE AND FALL OF MILK SECRETION WITH AGE.

Data are presented on the effect of age on milk secretion in the dairy cow. From the age when milk secretion usually begins (2 years) to the age when maximum body weight is reached (about 8 years) increase of milk secretion and increase of body weight with age follow the same exponential course, which is the course of a monomolecular reaction of chemistry. After this age, unlike body weight which remains practically constant, milk secretion declines exponentially, that is, the course of decline follows the course of decline of a monomolecular reaction. The whole course of milk secretion with age was therefore found to follow approximately the course of two simultaneous, consecutive, monomolecular reactions. This is taken to mean that growth and senescence go on simultaneously from the beginning to the end of life, and that each follows an exponential law with age; and therefore perhaps that the course of the two processes are limited by two consecutive chemical reactions.     

On the lag phase and initial decline of microbial growth curves

The lag phase is generally thought to be a period during which the cells adjust to a new environment before the onset of exponential growth. Characterizing the lag phase in microbial growth curves has importance in food sciences, environmental sciences, bioremediation and in understanding basic cellular processes. The goal of this work is to extend the analysis of cell growth curves and to better estimate the duration of the lag phase. A non-autonomous model is presented that includes actively duplicating cells and two subclasses of non-duplicating cells. The growth curves depend on the growth and death rate of these three subpopulations and on the initial proportion of each. A deterministic and a stochastic model are both developed and give the same results. A notable feature of the model is the decline of cells during the early stage of the growth curve, and the range of parameters when this decline occurs is identified. A limited growth model is also presented that accounts for the lag, exponential growth and stationary phase of microbial growth curves.     

THE ANALYSIS OF THE GROWTH OF THE NORMAL WHITE MOUSE INTO ITS CONSTITUENT PROCESSES

1. The several growth-cycles which are distinguishable in the growth of an animal or plant are mutually independent in that they do not share a common catalyser. 2. The growth of the white mouse has been shown to consist of three autocatalytic processes and one "linear process" of weight-accretion. The parameters of these processes have been evaluated for one strain and generation of mice. 3. The first and most extensive autocatalytic process is asymmetrical, being defined by an equation of the type: See PDF for Equation The second and third cycles, which are more rapid and do not begin to affect the growth of the animal until a later stage of development, are symmetrical, being defined by equations of the type: See PDF for Equation 4. The amplitude of the first autocatalytic growth-cycle in the mouse is almost the same in males and females, but the moment of maximum growth-velocity in the female anticipates that in the male, the velocity constant is smaller in the female, and the asymmetry estimated by the magnitude of the constant B, is greater in the female than in the male. 5. The amplitude of the second cycle is almost the same in males and females, but data are as yet lacking which would enable us to ascertain whether the velocity-constant and moments of maximum growth-velocity in this cycle differ in the two sexes or not. 6. The amplitude of the third cycle is much less in the female than in the male, and this difference of amplitude almost wholly accounts for the difference of adult weight in the two sexes. The velocity-constant of the third cycle is, however, greater in the female than in the male. Maximum growth velocity due to this cycle is attained at very nearly the same age in both sexes. 7. The origin of asymmetry in autocatalytic growth-processes is discussed. It is pointed out that asymmetry might originate in a progressive diminution of the velocity-constant. If this is the origin of the asymmetry of the first growth-cycle in the mouse, then it is shown that the velocity constant of autocatalysis in this cycle must be very nearly proportional to the nucleo-cytoplasmic ratio, as estimated by the chemical method of Le Breton and Schaeffer. 8. It is pointed out that no reliable measure of senescent loss of weight is available at present. It is shown that removal or decay of those conditions which initially maintain the separability of the growth-cycles which collectively constitute the growth of the white mouse would necessarily result in loss of weight.     

THE RATE OF GROWTH OF THE DAIRY COW : III. THE RELATION BETWEEN GROWTH IN WEIGHT AND INCREASE OF MILK SECRETION WITH AGE.

It is shown that from 2 years, the age when milk secretion usually begins, to 9 years, the age of maximum body weight, the increase of milk secretion with age follows the course of growth in body weight— both can be accurately represented by the equation of a monomolecular chemical reaction having a velocity constant of approximately the same numerical value. While increase in milk secretion and increase in body weight with age follow the same course, it is shown that increasing body weight contributes only about 20 per cent to increasing milk secretion with age. The fact that milk secretion and body weight follow the same course, even though they are largely independent of each other indicates that increase in body weight is a good measure of growth of the dairy cow; this fact also shows that the increase of milk secretion with age may be used as a measure of growth. The fact that milk secretion, like body weight, follows the course of a chemical reaction, adds further support to the theory that growth is limited by a chemical reaction.     

THE RATE OF SENESCENCE OF THE DOMESTIC FOWL AS MEASURED BY THE DECLINE IN EGG PRODUCTION WITH AGE

Data are presented showing that the course of decline of egg production with age in the domestic fowl from the time laying begins up to and including 8 years follows an exponential law, that is, each year''s egg production is a constant percentage of the preceding year''s production (88 per cent in the group of fowl studied). Since the exponential law is the same as the law of monomolecular change in chemistry, and since the course of egg production with age may be taken as an index of the course of senescence of organs, or tissues limiting egg production, it is suggested that this exponential law of egg production substantiates the idea that senescence is a physicochemical process the course of which is limited by a chemical reaction. It is shown that the exhaustion of the oocytes is not likely to be the factor limiting the course of egg production.     

THE RESPONSE OF GROWTH AND BIOCHEMICAL COMPOSITION TO VARIATIONS IN DAYLENGTH, TEMPERATURE, AND IRRADIANCE IN THE MARINE DIATOM THALASSIOSIRA PSEUDONANA (BACILLARIOPHYCEAE)

The relationships between the growth rate of the marine diatom Thalassiosira pseudonana (Hustedt) and irradiance, daylength, and temperature were determined in nutrient-sufficient semicontinuous cultures. The initial slopes of the growth versus total daily irradiance curves were not affected by temperature or daylength. Growth versus irradiance was best modeled as a hyperbolic function at short daylengths and better modeled as an exponential function at longer daylengths. The maximum or light-saturated growth rates at each daylength were modeled as a hyperbolic function of daylength. This model was extended in a novel manner to include temperature dependence providing a framework that can be used to interpret other experimental data on growth rate versus daylength. The resulting model should be useful in global models of phytoplankton growth. Carbon, nitrogen, and chl a quotas were influenced by daylength, irradiance, and temperature. Both C and N quotas were positive exponential functions of irradiance, whereas N and chl a quotas were significantly greater for cells grown at the lower temperature. The ratio chl a :C quota (chl a :Q_c) was a strong negative exponential function of total daily irradiance. Cells grown at 10° C had significantly greater chl a :Q_c ratios than those grown at 18° C, and daylength also had a significant positive influence on chl a :Q_c. The apparent effect of daylength on chl a :Q_c was removed by standardizing chl a :Q_c to growth rate (μ), resulting in a temperature-dependent relationship between chl a :Q_c·μ⁻¹ and irradiance that accounted for 95% of the variation in the data.     

温度对萼花臂尾轮虫卵的发育、种群增长和生产量的影响

1981—1983年,在不同的培养温度下,观察了萼花臂尾轮虫(Brachionus calyciflorus)卵的发育时间、种群的增长并用3种不同方法测算生产量。在5—30℃的培养温度下,轮虫卵的发育时间(D)随温度(T)升高而缩短,其曲线迴归方程为: LnD=2.0539+0.1097LnT-0.3046(LnT)2 在10,15,20,25℃的培养温度下,从休眠卵孵化出来的孤雌生殖雌体,其繁殖的种群增长曲线都呈“S”形,或称逻辑斯蒂曲线(Logistic curve)。不同的温度,种群达到高峰所需的时间有所不同,温度高者短,低者长;容纳量(carrying capacity)却随温度增加而有所增加。用线性和指数方法计算轮虫种群的生产量所得的结果相似;而与世代时间方法计算所获得的结果相比,差距很大。这种差距随着温度增加而增加。根据本文的研究结果和文献中报道的数据,获得了在0.6—35.2℃温度范围内,卵的发育时间(D)与温度(T)之间的迴归方程: LnD=2.1869-0.1919LnT-0.2218(LnT)2       

Impedance microbiology: quantification of bacterial content in milk by means of capacitance growth curves

The impedancimetric method is a technique for the rapid evaluation of milk bacterial content and also of its subproducts. Several authors have made use of culture conductance changes during bacterial growth for quantitative and qualitative assessments of microbial growth. However, interface capacitance curves, Ci, have not been used. In this paper, we quantify bacteria in cow raw milk by following their growth as the above-mentioned capacitance change time course event. With it, bigger growth variations, shorter detection times and a better coefficient of correlation with the plate count method were obtained than those yielded by conductance curves. Calibration was performed by plotting initial known concentrations, IC (CFU/ml), as a function of the time detection theshold (TDT).     

Non-exponential growth by mammalian cells in culture

The concept of exponential growth by mammalian cells in culture is based upon the apparent linearity of semilogarithmic data plots. This method of graphical analysis is known to be an unreliable test of the exponential hypothesis. We have re-examined the question of growth exponentiality using the more sensitive method of Smith plots, in which specific growth rate is plotted against either time or density on transformed graphical coordinates which linearize the mathematical expression of the growth hypothesis being tested. With exponential growth, data points fall on a horizontal straight line when specific growth rate is plotted against time or density. Using both our own and literature data, we have performed Smith plot analyses on the growth of 125 different mammalian and avian cell lines. Of these, only eleven exhibited an exponential phase. The remaining cell lines all had non-exponential growth patterns. The most common of these consisted of an initial period of growth acceleration followed by a later phase of deceleratory growth. A smaller number of lines exhibited deceleratory kinetics at all times after plating. We conclude that mammalian cell growth in culture is predominantly non-exponential, and that the apparent exponentiality of semilogarithmic data plots is usually a methodological artifact.     

THE LOCI OF REPEATED EVOLUTION: A CATALOG OF GENETIC HOTSPOTS OF PHENOTYPIC VARIATION

What is the nature of the genetic changes underlying phenotypic evolution? We have catalogued 1008 alleles described in the literature that cause phenotypic differences among animals, plants, and yeasts. Surprisingly, evolution of similar traits in distinct lineages often involves mutations in the same gene (“gene reuse”). This compilation yields three important qualitative implications about repeated evolution. First, the apparent evolution of similar traits by gene reuse can be traced back to two alternatives, either several independent causative mutations or a single original mutational event followed by sorting processes. Second, hotspots of evolution—defined as the repeated occurrence of de novo mutations at orthologous loci and causing similar phenotypic variation—are omnipresent in the literature with more than 100 examples covering various levels of analysis, including numerous gain‐of‐function events. Finally, several alleles of large effect have been shown to result from the aggregation of multiple small‐effect mutations at the same hotspot locus, thus reconciling micromutationist theories of adaptation with the empirical observation of large‐effect variants. Although data heterogeneity and experimental biases prevented us from extracting quantitative trends, our synthesis highlights the existence of genetic paths of least resistance leading to viable evolutionary change.     

Non-Exponential Growth By Mammalian Cells In Culture

The concept of exponential growth by mammalian cells in culture is based upon the apparent linearity of semilogarithmic data plots. This method of graphical analysis is known to be an unreliable test of the exponential hypothesis. We have re-examined the question of growth exponentiality using the more sensitive method of Smith plots, in which specific growth rate is plotted against either time or density on transformed graphical coordinates which linearize the mathematical expression of the growth hypothesis being tested. With exponential growth, data points fall on a horizontal straight line when specific growth rate is plotted against time or density. Using both our own and literature data, we have performed Smith plot analyses on the growth of 125 different mammalian and avian cell lines. of these, only eleven exhibited an exponential phase. the remaining cell lines all had non-exponential growth patterns. the most common of these consisted of an initial period of growth acceleration followed by a later phase of deceleratory growth. A smaller number of lines exhibited deceleratory kinetics at all times after plating. We conclude that mammalian cell growth in culture is predominantly non-exponential, and that the apparent exponentiality of semilogarithmic data plots is usually a methodological artifact.     

THE SIGNIFICANCE OF THE HYDROGEN ION CONCENTRATION FOR THE DIGESTION OF PROTEINS BY PEPSIN

The experiments described above show that the rate of digestion and the conductivity of protein solutions are very closely parallel. If the isoelectric point of a protein is at a lower hydrogen ion concentration than that of another, the conductivity and also the rate of digestion of the first protein extends further to the alkaline side. The optimum hydrogen ion concentration for the rate of digestion and the degree of ionization (conductivity) of gelatin solutions is the same, and the curves for the ionization and rate of digestion as plotted against the pH are nearly parallel throughout. The addition of a salt with the same anion as the acid to a solution of protein already containing the optimum amount of the acid has the same depressing effect on the digestion as has the addition of the equivalent amount of acid. These facts are in quantitative agreement with the hypothesis that the determining factor in the digestion of proteins by pepsin is the amount of ionized protein present in the solution. It was shown in a previous paper that this would also account for the peculiar relation between the rate of digestion and the concentration of protein. The amount of ionized protein in the solution depends on the amount of salt formed between the protein (a weak base) and the acid. This quantity, in turn, according to the hydrolysis theory of the salts of weak bases and strong acids, is a function of the hydrogen ion concentration, up to the point at which all the protein is combined with the acid as a salt. This point is the optimum hydrogen ion concentration for digestion, since the solution now contains the maximum concentration of protein ions. The hydrogen ion concentration in this range therefore is merely a convenient indicator of the amount of ionized protein present in the solution and takes no active part in the hydrolysis. After sufficient acid has been added to combine with all the protein, i.e. at pH of about 2.0, the further addition of acid serves to depress the ionization of the protein salt by increasing the concentration of the common anion. The hydrogen ion concentration is, therefore, no longer an indicator of the amount of ionized protein present, since this quantity is now determined by the anion concentration. Hence on the acid side of the optimum the addition of the same concentration of anion should have the same influence on the rate of digestion irrespective of whether it is combined with hydrogen or some other ion (provided, of course, that there is no other secondary effect of the other ion). The proposed mechanism is very similar to that suggested by Stieglitz and his coworkers for the hydrolysis of the imido esters. Pekelharing and Ringer have shown that pure pepsin in acid solution is always negatively charged; i.e., it is an anion. The experiments described above show further that it behaves just as would be expected of any anion in the presence of a salt containing the protein ion as the cation and as has been shown by Loeb to be the case with inorganic anions. Nothing has been said in regard to the quantitative agreement between the increasing amounts of ionized protein found in the solution (as shown by the conductivity values) and the amount predicted by the hydrolysis theory of the formation of salts of weak bases and strong acids. There is little doubt that the values are in qualitative agreement with such a theory. In order to make a quantitative comparison, however, it would be necessary to know the ionization constant of the protein and of the protein salt and also the number of hydroxyl (or amino) groups in the protein molecule as well as the molecular weight of the protein. Since these values are not known with any degree of certainty there appears to be no value at present in attempting to apply the hydrolysis equations to the data obtained. It it clear that the hypothesis as outlined above for the hydrolysis of proteins by pepsin cannot be extended directly to enzymes in general, since in many cases the substrate is not known to exist in an ionized condition at all. It is possible, however, that ionization is really present or that the equilibrium instead of being ionic is between two tautomeric forms of the substrate, only one of which is attacked by the enzyme. Furthermore, it is clear that even in the case of proteins there are difficulties in the way since the pepsin obtained from young animals, or a similar enzyme preparation from yeast or other microorganisms, is said to have a different optimum hydrogen ion concentration than that found for the pepsin used in these experiments. The activity of these enzyme preparations therefore would not be found to depend on the ionization of the protein. It is possible of course that the enzyme preparations mentioned may contain several proteolytic enzymes and that the action observed is a combination of the action of several enzymes. Dernby has shown that this is a very probable explanation of the action of the autolytic enzymes. The optimum hydrogen ion concentration for the activity of the pepsin used in these experiments agrees very closely with that found by Ringer for pepsin prepared by him directly from gastric juice and very carefully purified. Ringer''s pepsin probably represents as pure an enzyme preparation as it is possible to prepare. There is every reason to suppose therefore that the enzyme used in this work was not a mixture of several enzymes.     

THE RATE OF DECLINE OF MILK SECRETION WITH THE ADVANCE OF THE PERIOD OF LACTATION

It is shown that the course of decline of milk secretion with the advance of the period of lactation may be expressed by the equation of a monomolecular chemical reaction; that is, the percentage decline of milk secretion with the advance of the stage of lactation is constant. This substantiates the idea that milk secretion is limited by a chemical reaction, and, in general, brings lactation into the class of processes the speed of which is determined by the concentration of a limiting substance.