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.     

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 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.     

A NOTE ON THE SIMILARITIES BETWEEN THE CURVES OF GROWTH AND OF REGENERATION

The curves of growth and of regeneration follow the same course, and can be represented by the same exponential equation. This is taken to substantiate the theory that growth and regeneration are essentially identical processes governed by the same laws. A common peculiarity of the curves of growth and of regeneration is that during a short period in the early stages of regeneration and of growth, the apparent observed speed of these processes seems to be relatively slow. As a result, the curve of the fitted equation cuts the time axis not at zero, the beginning of growth or regeneration, but somewhat later. Data on regeneration are cited indicating that the initial slow phase of regeneration is due to the time required for the formation of a cap of embryonic cells which serves as a basis for the more active later regeneration; in other words, to qualitative growth which cannot be expressed in terms of quantitative units. It is suggested that the apparent initial slow phase of growth of the individual from the fertilized egg is due to a similar qualitative growth. It is suggested that if the initial qualitative changes could be converted into some common unit with the subsequent quantitative changes, the apparent initial lag would disappear, and the exponential equation representing the course of these processes would then be the same as the equation used to represent the course of a monomolecular chemical reaction. Certain implications of this reasoning are discussed in the text.     

Evolution of senescence: Influence of age-dependent death rates on the natural increase of a hypothetical population

A hypothetical population is characterized by functions of age which describe its longevity and its maternity rate. Solution of the renewal equation for the birth rate of the population yields a characteristic equation which, in contradiction to the results of previous studies, may have more than one real root. The largest real root of the characteristic equation is the rate of natural increase, r, of the population and is used as a measure of its selective advantage.The maternity rate is represented by a rising or falling exponential function of age. Longevity is represented by a series each term of which has the form of a gamma distribution function. As the number of terms increases, the mean longevity remains constant, but the function becomes progressively more rectangular in shape; the early death rate declines, while the death rate in old age increases. Unless the reproductive fraction is small, each such decrease in the youthful death rate more than compensates for the corresponding increase in old age and causes an increment in r which is interpreted as a step toward the evolution of senescence. Although the degree of change in r attendant upon a change in the age-dependency of the death rate is related to the initial value of the maternity function, it is not influenced by the age dependency of the maternity function.     

THE RATE OF GROWTH OF THE DAIRY COW : EXTRAUTERINE GROWTH IN WEIGHT.

The growth period of the Jersey and Holstein cows is made up of at least three cycles, two extrauterine cycles with maxima at about 5 and 20 months of age, and one intrauterine cycle, the maximum of which has not yet been determined. The equation of an autocatalytic monomolecular reaction was found to give very good results when applied to the cycle having its maximum at about 5 months of age. The values obtained from this equation when applied to the cycle having the maximum at about 20 months of age were higher than the observed values probably due to the retarding effect of pregnancy and lactation on growth.     

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.     

Parallel lines: nothing has changed?

The exponential increase in mortality rate with age is a universal feature of aging and is described mathematically by the Gompertz equation. When this equation is transformed semilogarithmically, it conforms to a straight line, the slope of which is generally used to reflect the rate of senescence. Historical and contemporary data of human and nonhuman populations show that adverse environmental conditions do not always change the slope of the log mortality rate over age. From these latter observations it is sometimes mistakenly inferred that the rate of senescence is unaffected by environmental conditions. Current biological inference emphasizes that gene action is dependent on the environment in which it is expressed. Here, we propose using the tangent line of the Gompertz equation to assess whether the rate of senescence has altered. Such an approach unmasks different rates of senescence when parameter G has remained constant, an observation that is in line with the notion that a plastic life history trait such as the rate of senescence results from the interplay of both genes and environment.     

The cost of growing large: costs of post‐weaning growth on body mass senescence in a wild mammal

Individual body mass often positively correlates with survival and reproductive success, whereas fitness costs of growing large are rarely detected in vertebrates in the wild. Evidence that adult body mass progressively declines with increasing age is accumulating across mammalian populations. Growing fast to a large body can increase the cellular damage accumulated throughout life, leading body growth in early life to be negatively associated with the rate of body mass senescence. Moreover, the onset of mass senescence may strongly depend on both sex‐specific reproductive tactics and environmental conditions. Assessing the timing and the rate of body mass decline with increasing age thus offers an opportunity to look for costs of having grown fast, especially after a poor start during early life, in both sexes and in different environments. Using a unique dataset including 30 years of longitudinal data on age‐specific body mass collected in two roe deer Capreolus capreolus populations subjected to contrasted environmental conditions, we looked for potential costs of high post‐weaning growth rate in terms of steeper rate of body mass senescence. Our analyses of body mass senescence accounted for the potential variation in the onset of senescence and allowed explicit comparisons of this variable between sexes and populations. Higher growth rates late in the growing period (after weaning) were associated with a steeper rate of body mass senescence, regardless of early mass (gained before weaning), but at different extents depending on sex and environmental conditions. Body mass senescence occurred earlier in males than in females, especially in the population facing limiting resources. In the wild, although heavy individuals generally survive better than small ones, the costs of growing large late in the growing period only became apparent late in life through mass senescence.     

Density dependence triggers runaway selection of reduced senescence

In the presence of exogenous mortality risks, future reproduction by an individual is worth less than present reproduction to its fitness. Senescent aging thus results inevitably from transferring net fertility into younger ages. Some long-lived organisms appear to defy theory, however, presenting negligible senescence (e.g., hydra) and extended lifespans (e.g., Bristlecone Pine). Here, we investigate the possibility that the onset of vitality loss can be delayed indefinitely, even accepting the abundant evidence that reproduction is intrinsically costly to survival. For an environment with constant hazard, we establish that natural selection itself contributes to increasing density-dependent recruitment losses. We then develop a generalized model of accelerating vitality loss for analyzing fitness optima as a tradeoff between compression and spread in the age profile of net fertility. Across a realistic spectrum of senescent age profiles, density regulation of recruitment can trigger runaway selection for ever-reducing senescence. This novel prediction applies without requirement for special life-history characteristics such as indeterminate somatic growth or increasing fecundity with age. The evolution of nonsenescence from senescence is robust to the presence of exogenous adult mortality, which tends instead to increase the age-independent component of vitality loss. We simulate examples of runaway selection leading to negligible senescence and even intrinsic immortality.     

Senescence of detached leaves in pigeon pea and chick pea: Regulation by developing pods

The senescence rate of the subtending leaves in deflowered and control plants of pigeon pea ICajanus cajan (L.) Millsp. cv., Prabhat] and chick pea ( Cicer arietinum L. cv. JG 62) were examined during the course of natural and induced senescence, at several stages of pod growth. The leaves from the top 5 nodes on the main axis in pigeon pea and the top 8 nodes on the main axis in chick pea were used throughout the experiments. The natural senescence was characterized in leaves taken directly from the field-growing plants. For the study of induced senescence, the leaves were excised from both control and deflowered plants at various stages of pod growth and placed in test tubes containing water under dark conditions. Senescence was assessed in terms of peroxidase activity and contents of tola] chlorophyll, soluble amino acids and total protein. During natural ageing in the field, the leaves from deflowered plants exhibited delayed senescence in both the species. In contrast, the rate of ageing during induced senescence was higher in the leaves of deflowered plants than in the controls. Although of the same chronological age when excised for induced senescence, the leaves of deflowered plants were evideatly metabolically different from the controls, due to the fact that deflowered plants did not support the development of pods. This difference probably determined the subsequent rate of induced senescence.     

Adiabatic solution of the Ohmic cable equation. General theory, homogeneous cylindrical fiber]

An adiabatic solution of the Ohmic cable equation is suggested, which reduces the non-stationary equation to a stationary form. The adiabatic length constant of the stationary equation is time-dependent. The adiabatic solutions for the boundary conditions that change in time linearly and exponentially were studied. In the latter case, the adiabatic length constant does not depend on time though it differs from the usual length constant. The cable input characteristics of exact and adiabatic solutions were compared in the cases of the voltage- and current-clamp, and electric field stimulation. The adiabatic and exact solutions are identical for the rising exponential stimuli. For the falling exponential stimuli, the adiabatic solution determines the exact asymptotic solution if the stimulus decays slower than the relaxation of initial conditions. It is propose to use linear and exponential ramp stimulation in electrotonic measurements.     

THE RATE OF GROWTH OF THE DAIRY COW : II. GROWTH IN WEIGHT AFTER THE AGE OF TWO YEARS.

An extensive amount of data is presented on the growth in weight of the dairy cow from 2 to 17 years of age, covering practically the entire duration of life. The data show that after the age of 2 years the rate of growth declines in a non-cyclic manner. The course of decline in growth follows the course of decline of a monomolecular chemical reaction; that is, the percentage decline in growth with age is constant.     

Evolution of senescence: Natural increase of populations displaying gompertz- or power-law death rates and constant or age-dependent maternity rates

Several hypothetical populations which differ in degrees of senescence are compared with respect to their rates of natural increase. The rate of natural increase is employed as a measure of selective advantage. The populations are characterized by their maternity and death rates, expressed as functions of age. Maternity rates are described by constant or quasi-human, age-dependent functions. Death rates are described by constant, Gompertzian (exponential) or power functions. Longevity functions, representing the probability of survival to a specific age, are obtained by integrating the death rate functions. The degree of senescence of a population is measured by the rapidity of ascent of its death-rate function or by the rectangularity of its longevity function. The increase in death rate late in life which constitutes senescence is compensated by a decrease in death rate early in life. The balance between the two changes in rate is, by assumption, such that the mean value of the longevity function is independent of the degree of senescence. This assumption makes it possible to separate the effects produced by the evolution of senescence from those caused by changes in longevity.The rate of natural increase is obtained by numerical solution of an integral characteristic equation. The results show that senescence is advantageous in all populations except those in which the maternity function is constant and the size is declining at a rapid rate. When the parameters entering into the longevity functions have values such that the functions approximate human longevity data, the improvement in the rate of natural increase resulting from senescence closely approaches limiting values obtained with the use of a precisely rectangular longevity function. Other results support the observation that reproduction at an early age confers greater selective advantage than equivalent reproduction later in life.     

Growth of Ca-D-malate crystals in a bioreactor

To develop a bioreactor for solid-to-solid conversions, the conversion of solid Ca-maleate to solid Ca-D-malate by permeabilized Pseudomonas pseudoalcaligenes was studied. In a bioreactor seeded with product (Ca-D-malate) crystals, growth of Ca-D-malate crystals is the last step in the solid-to-solid conversion and is described here. Crystal growth is described as a transport process followed by surface processes. In contrast to the linear rate law obeyed by the transport process, the surface processes of a crystal-growth process can also obey a parabolic or exponential rate law. Growth of Ca-D-malate crystals from a supersaturated aqueous solution was found to be surface-controlled and obeyed an exponential rate law. Based on this rate law, a kinetic model was developed which describes the decrease in supersaturation due to Ca-D-malate crystal growth as a function of the constituent ions, Ca(2+) and D-malate(2-). The kinetic parameters depended on temperature, but, as expected (surface-controlled), they were hardly affected by the stirring speed.     

Control of Leaf Growth and its Role in Determining Variation in Plant Growth Rate from an Ecological Perspective

Abstract: Plants vary widely in their relative growth rate (RGR), be it dependent on environmental conditions or due to their genetic background. In a comparison of the RGR of grasses growing under different environmental conditions, variation in RGR tends to correlate with that in the leaf elongation rate (LER). When different species or genotypes thereof are compared under identical growing conditions, variation in LER may or may not correlate with that in RGR, depending on the comparison. However, since RGR is described by an exponential equation, whereas LER is mainly a linear process, we conclude that any correlation between RGR and LER must be fortuitous. That is, exponential growth must be due to increases with time in plant traits such as 1) leaf dry mass per unit leaf length invested per unit time, and/or 2), i.e., the total LER of all the growing leaves at one point in time. The latter can be achieved as follows: 1) each subsequent leaf has a higher LER than the preceding one; 2) leaves appear at an increasing rate; 3) the duration of the process of leaf elongation increases for subsequent leaves. In this review, we only explore possible factors that account for changes in with time, in different genotypes and under different environmental conditions. Inherent variation in LER of individual leaves and variation due to environmental factors may reflect variation in the rate of cell division and/or in cell elongation.     

Chemostat and in‐situ colonization kinetics of thermothrix thiopara on calcite and pyrite surfaces

Growth and attachment rates of Thermothrix thiopara on calcite and pyrite were quantitated in a thiosulfate‐limited chemostat and in the thermal spring where the organism is found in nature. Surface growth rates were quantitated by using the surface colonization and exponential growth equations. These two models were compared as means of determining surface growth rates. In the chemostat, T. thiopara cells colonizing calcite and pyrite surfaces grew at approximately one‐third the rate of suspended cells. However, T. thiopara attached to pyrite faster than to calcite. In the thermal spring, growth and attachment rates were equal on calcite and pyrite. It was concluded that the exponential growth equation overestimates in‐situ surface growth rates and that T. thiopara grows more slowly when colonizing mineral surfaces than when growing in suspension. Lower growth rates on surfaces may be due to a reduced cell surface area for nutrient uptake or an increased specific maintenance rate.     

Cell population dynamics during the differentiative phase of tissue development

The dynamics of a cell population whose numbers are growing exponentially have been described well by a mathematical model based on the theory of age-dependent branching processes. Such a model, however, does not cover the period following exponential growth when cell differentiation curtails population size. This paper offers an extension to the branching process model to remedy this deficiency. The extended model is ideal for describing embryonic growth; its use is illustrated with data from embryonic retina. The model offers a better computational framework for the interpretation of a variety of data (growth curves of cell numbers, DNA histograms, thymidine labelling indices, FLM curves, BUdR-labelled mitoses curves) because age-distributions can be calculated at any stage of development, not just during exponential growth. Proportions of cells in the various phases of the cell cycle can be computed as growth slows. Such calculations show the gradual transition from a population dominated by cells which are young with respect to cell cycle age to one dominated by those which are old, and the effects such biases have on the proportions of cells in each phase.     

A biomathematical model of a human operator’s falling asleep

Quantitative analysis of the transition from wakefulness to sleep and prediction of the moment when errors in professional activity appear because of a decrease in the arousal level require microinterval monitoring of falling asleep. A psychomotor test was developed that rapidly decreased the arousal level, which made it possible to record as many as 10–20 episodes of correct and erroneous activity within 40 min and isolate the periods electrophysiologically corresponding to wakefulness and brief sleep. Seventy subjects were tested, and 6700 fragments of recordings with correct and erroneous performance were analyzed. Analysis of the experimental data showed that the transition from wakefulness to sleep includes intermediate short and relatively long periods of wakefulness and sleep, whose durations are distributed according to the double exponential law. A mathematical model describing the time course of alternation of these four states of wakefulness and sleep predicts the probability of prolonged, potentially dangerous disturbances in operator activity because of microsleep as dependent on the initial state and individual characteristics of subjects. The results will be useful both for the development of devices monitoring and predicting changes in the physiological arousal level and for analysis of traffic and industrial accidents.