Mortality-rate crossovers and maximum lifespan in advantaged and disadvantaged populations: accelerated-mortality and sudden-death models.

One population is advantaged relative to another by our definition if its survival function is greater at all ages. A population has a lifespan maximum if there is an age at which its survival function becomes exactly zero. Earlier work concerned conditions under which the mortality-rate functions of advantaged and disadvantaged populations displaying lifespan maxima always crossed. Here two survival models of populations having lifespan maxima are presented in which mortality-rate crossings between advantaged and disadvantaged subpopulations may fail to appear. One, the accelerated-mortality model, has a continuous survival function; in the other, the sudden-death model, the survival function is discontinuous. Both differ from examples examined previously in that their mortality-rate functions become infinite at their lifespan maxima.

Intersections of mortality-rate and survival functions: model-independent considerations.

In work reported previously (Hirsch, 1995), it was shown that families of straight lines intersect at a single point if and only if the slopes of the lines are linearly related to their intercepts. This slope-intercept relation was applied to several mathematical mortality models including the Gompertz-Makeham and the Weibull. In all cases, survival functions intersected at greater ages than the corresponding mortality-rate functions. It was further demonstrated that a common point of intersection can exist for members of a family of survival functions or for members of the corresponding family of mortality-rate functions but not for both. Here the same results are obtained with respect to intersections of general model-independent survival and mortality-rate functions. The generality of the results strengthens the conclusion reached earlier that these intersections imply only the existence of a valid slope-intercept relation and have little other significance with regard to the biology of aging.

Do intersections of mortality-rate and survival functions have significance?

Common points of intersections have frequently been reported among members of families of linearized mortality-rate and survival functions. A general condition for the existence of such intersections is derived. It is shown that a common point of intersection between straight-line functions exists if and only if the intercepts of the functions are linearly related to their slopes. This slope-intercept condition is applied to a didactic model to illustrate its generality and to three models, the Gompertz-Makeham, the Weibull, and the logistic, which are often used in the analysis of mortality data. The slope-intercept condition for the Gompertz-Makeham mortality-rate model proves to be the well-known Strehler-Mildvan correlation. Families of mortality-rate functions or of the corresponding survival functions but not both may display common points of intersection. Differences between the ages at which survival functions intersect and those at which the associated mortality-rate functions intersect are calculated to be of the order of magnitude of 10 to 20 years. Survival function intersections lie close to the limit of human life span but often arise in consequence of unsupported extrapolations of data obtained at younger ages. These and other results lead to the conclusion that, in themselves, the intersections of survival and mortality-rate functions are not of great importance. To the extent that significance can be attributed to the intersections, it lies in the existence of linear relationships between their slopes and intercepts.

Can an improved environment cause maximum lifespan to decrease? Comments on lifespan criteria and longitudinal Gompertzian analysis.

Longitudinal Gompertzian analysis yields the counterintuitive conclusion that an improved environment can cause a decrease in maximum lifespan. The basis for this conclusion is examined. Results include the following: 1) The use of a specified high mortality rate as a criterion for maximum lifespan is arbitrary and leads to a calculated lifespan which is quite sensitive to the value of the criterion. 2) The definition of lifespan as the age to which a specified small population fraction survives is less arbitrary and less sensitive to the chosen criterion value. 3) However, the use of a survival criterion for lifespan in place of a mortality-rate criterion does not eliminate the seeming contradiction between environmental improvement and decreased lifespan. 4) Mortality rates can be approximated in semilogarithmic coordinates by three straight-line segments. The first segment, applicable through age 85, is the conventional Gompertz function. The second segment, representing ages 85 through 96, has a lower slope than the first, while the third segment, representing ages 96 through 124, has a negative slope. 5) The mortality rate obtained by extrapolating the first segment to a nominal age of maximum lifespan differs markedly from the true mortality rate at that age. 6) The conclusion that an improved environment is associated with a reduction in lifespan arises as a consequence of such an extrapolation.

Accumulation of a senescence factor in yeast cells.

A mathematical model is used to describe the increase in generation time with age of mother cells in the asymmetrically dividing yeast Saccharomyces cerevisiae. It is postulated that the generation time increases linearly with the amount of a senescence factor present in each cell. The senescence factor, which inhibits cell division is produced continuously. Although it is not degraded or destroyed, it may be diluted by cell division. The model is formally identical to one which was successful in describing waste dilution in populations of aging human diploid fibroblasts (Hirsch, 1978). The relation between aging in yeast and in fibroblasts is explored. Agreement between calculated results and yeast cell generation-time data is adequate. The results indicate that the senescence factor accumulates with little or no dilution in aging yeast mother cells.

The waste-product theory of aging: simulation of metabolic waste production.

A mathematical model of cellular metabolism is used to relate the rates of cell division and waste production to the concentrations of oxygen and glucose in the medium in which a normal diploid cell culture is grown. The metabolic model in tandem with an earlier waste-content model based on the waste-product theory of aging provides a unified cell-culture model with which population size and intracellular waste content can be calculated. Population size is measured by the number of population doublings which have been achieved. After suitable adjustment of parameters in the metabolic model, maximum values of population size are calculated numerically with the use of the unified model. Results show that the population maxima are related in a plausible way to the oxygen and glucose concentrations. The effects of temperature changes and contact inhibition of growth are also simulated. Small changes in the cell-division and waste-production rates can cause transformation to unlimited growth in the waste-content model, but the unified model is not correspondingly sensitive to changes in the oxygen and glucose concentrations or to changes in temperature.

The waste-product theory of aging: transformation to unlimited growth in cell cultures.

A differential equation governing intracellular waste content is solved numerically to determine the circumstances under which the growth of an in vitro cell population is limited. Parameter values derived from data on human glial cell cultures are employed. It is assumed that a) waste accumulation depresses the rate of cellular reproduction and b) intracellular waste is diluted by cell division, but is not otherwise eliminated. Population size depends upon two parameters: the rate of waste production and the rate of cell division in the absence of waste. If the rate of waste production is sufficient, the population size approaches an asymptote as in phase III growth in vitro. If a lower rate of waste production allows the cells to outmultiply the waste, growth is unlimited as in a transformed cell population. The asymptotic population size and the threshold for unlimited growth are remarkably sensitive to small changes in the values of the two rate parameters unless the ratio of their values is constant. This suggests that there may be a cellular mechanism that relates the waste production and cell division rates.

Why should senescence evolve? An answer based on a simple demographic model.

The demographic model of senescence described here provides an answer to the question, "Why should senescence evolve?" Most generally stated, the answer is that senescence should be expected to evolve if its negative effect on the rate of natural increase of a nonsenescent population is sufficiently offset by the early appearance of an advantageous characteristic. This is a nonadaptive point of view in the sense discussed by Kirkwood (1985) and by Kirkwood and Cremer (1982). It corresponds more closely to Medawar's (1952) position than to Weisman's (1889). The demographically based model in which senescence is represented by sudden death supplies an explanation which is simple and credible for the evolution of senescence. It supports the following specific conclusions: 1. The introduction of sudden death (case 2) in a nonsenescent population otherwise subject only to randomly occurring death (case 1) is, by itself, disadvantageous from the standpoint of natural selection. 2. However the population may enjoy a net selective advantage if the disadvantage of sudden-death senescence is compensated by an appropriate improvement early in its life history, e.g., by a reduction in its presenescent death rate. 3. In the most extreme example possible, in which the presenescent death rate is zero and the survival curve is rectangular (case 3), the early improvement is associated with an increase in the degree of sensescence of the population, in its mean longevity, and in its average age at death. Thus natural selection can simultaneously favor both senescence and longevity. 4. Among populations in which the age of sudden death is balanced against the presenescent death rate in such a way that mean longevity is held constant (case 4), sudden-death senescence provides selective advantage relative to a nonsenescent population (case 1). Up to the point at which the survival curve becomes rectangular, the earlier the age at which sudden-death occurs, the greater the selective advantage. Similar conclusions were reached earlier with respect to forms of senescence which take effect more gradually than the sudden-death mechanism postulated here. 5. Populations in which the age of sudden death is balanced against the presenescent death rate in such a way that the average age at death is held constant (case 5) are selectively neutral with respect to a nonsenescent population (case 1). Thus a reduction in mortality at a nearly age can compensate for the sudden death of the whole population at an advanced age because so few individuals in the nonsenescent population survive to reproduce when old.

The waste-product theory of aging: cell division rate as a function of waste volume.

The rate of cell division is calculated as a function of waste product volume in U-787CG human diploid glial cells grown in vitro. The calculation is based on two earlier mathematical models. One is a compartmental analysis in which cell division rate is obtained from data on the fraction of cells which become sterile as the passage level increases. A second model is used to calculate the amount of waste per cell from the observed rate of waste accumulation in a non-dividing population and from the division rate calculated with the use of the first model. Results from the two models are correlated to obtain the desired function relating cell division rate to waste volume. If cellular aging is taken to mean loss of the ability of cells to divide, and if, as in the waste-product theory, this loss is attributed to waste accumulation, the calculated results show that aging is evident at waste levels well below those at which non-dividing populations can survive. Thus the process of cell division may be much more sensitive to waste accumulation than other cellular processes needed for the maintenance of life.

Influence of the existence of a resting state on the decay of synchronization in cell culture.

General relationships between the distribution of cell doubling times and the growth pattern of an initially synchronized cell population are applied to the model proposed by Smith and Martin (1973) in which the mitotic cycle or "B" phase is preceded by a random-exit resting "A" state. Results show that culture synchronization decays so rapidly as to be virtually unobservable unless the time spent by a cell in the B phase is at least equal to that spent in the A state. If synchronization persists over several mitotic cycles, the growth pattern is determined to a much greater extent by variation in the duration of the B phase than by the probability of exit from the A state. Accordingly the growth pattern of a cell population, like the doubling time distribution which governs the pattern, is of limited usefulness in detecting the existence of a resting state.

Survival and aging of a small laboratory population of a marine mollusc, Aplysia californica.

In an investigation of the postmetamorphic survival of a population of 112 Aplysia californica, five animals died before 100 days of age and five after 200 days. The number of survivors among the 102 animals which died between 100 and 220 days declined approximately linearly with age. The median age at death was 155 days. The animals studied were those that died of natural causes within a laboratory population that was established to provide Aplysia for sacrifice in an experimental program. Actuarial separation of the former group from the latter was justified by theoretical consideration. Age-specific mortality rates were calculated from the survival data. Statistical fluctuation arising from the small size of the population was reduced by grouping the data in bins of unequal age duration. The durations were specified such that each bin contained approximately the same number of data points. An algorithm for choosing the number of data bins was based on the requirement that the precision with which the age of a group is determined should equal the precision with which the number of deaths in the groups is known. The Gompertz and power laws of mortality were fitted to the age-specific mortality-rate data with equally good results. The positive values of slope associated with the mortality-rate functions as well as the linear shape of the curve of survival provide actuarial evidence that Aplysia age. Since Aplysia grow linearly without approaching a limiting size, the existence of senescence indicates especially clearly the falsity of Bidder's hypothesis that aging is a by-product of the cessation of growth.

Influence of the existence of a resting state on the probability of cell division in culture.

A cell cycle model developed by Smith and Martin is generalized to allow for the possibility that the duration of the B phase is not fixed. The B phase is the equivalent of the traditional S, G2, and M phases of the cell cycle. The duration of the B phase is represented by a Gaussian probability distribution; the duration of the resting or A state which replaces the traditional G1 phase is represented by a decaying exponential distribution. A doubling time distribution, termed the CEG distribution, is obtained by convolution of the A state and B phase distributions. Like the reciprocal normal, rate normal, and log normal distributions, it is a rounded unimodal peak that is skewed to the right. None of the three former distributions is associated with a cell cycle model that includes a resting state. However the CEG distribution, which is so associated, bears little resemblance to the delayed exponential distribution which results when the duration of the B phase is fixed and the duration of the A state is random. Consequently, it would be difficult to use the doubling time distribution to determine whether or not a resting state exists in a particular cell population.

Phase-space description of the cell cycle: application to noncycling, senescent, and transformed cells.

The behavior of a cellular biochemical reaction system can be portrayed by its trajectory in phase space. Phase-plane trajectories are proposed which depict the concentration of a characteristic chemical species in normally reproducing diploid cells, and in their noncycling, transformed, and senescent counterparts. It is assumed that the biochemical reaction system which determines the concentration of the characteristic species is nonlinear and nonconservative; the trajectories are therefore analyzed with the help of established qualitative mathematical techniques which are applicable to such systems. Mitotic cycles are presented as stable limit cycles; noncycling and senescent states are represented as isolated stable singular points. The concentration trajectory traversed in the transformed cell cycle surrounds the trajectory traversed in the normal cell cycle, which itself surrounds the singular point corresponding to the noncycling or senescent state. Transitions between the normal cycle and the noncycling state are associated with changes in the concentration of the characteristic species which exceed cyclically varying rate-dependent threshold values. Transitions from the normal cycle to the transformed cycle or to the senescent state are described by the coalescence and disappearance of the normal stable limit cycle with one of two adjacent unstable limit cycles.

Commitment theory of cellular aging: possibility of an immortal diploid cell strain.

Calculations based on the commitment theory of cellular aging indicate that the mean number of uncommitted cells is much greater than unity in populations consisting of 10(10) or more diploid cells. Consequently the probability that all of the uncommitted cells will be lost by in vitro subcultivation is very small. It may therefore be possible to establish a cell population in the laboratory which is, for all practical purposes, immortal.

The waste-product theory of aging: waste dilution by cell division.

When cells divide, the quantity of waste material per cell decreases because the wastes are "diluted" by apportionment between the daughters which result from the division. The quantity of waste present in a symmetrically or asymmetrically dividing population of cells is governed by a first-order non-linear differential equation. In the derivation of the equation, it is assumed (a) that waste is created at a rate which is either constant or proportional to the amount of waste already formed, (b) that waste is neither destroyed nor transported across cell walls, and (c) that the rate of cell division at large values of time is inversely proportional to the amount of waste per cell raised to a power. Relations among the parameters of the differential equation specify conditions under which its solutions rise to a critical value. If the amount of waste per cell given by a solution of the differential equation exceeds this value, it is assumed that deleterious effects become evident and that cell death follows. Decreases in the cell division rate leading to a cessation of population growth may occur at lower levels of waste accumulation.

Responses of single units in the cat cochlear nucleus to sinusoidal amplitude modulation of tones and noise: linearity and relation to speech perception.

Responses to the sinusoidal modulation envelopes of amplitude-modulated tonend noise carriers were recorded from single units in the cochlear nucleus of the cat. The unit discharges were synchronized to the peaks of the modulation envelope. Population-averaged firing rate were independent of the modulation index of the stimulus. Temporal firing patterns, as represented by the shapes and magnitudes of modulation-cycle histograms, were strongly dependent on stimulus intesity and modulation index. Nonlinear nonsinusoidal responses to sinusoidal modulation envelopes were observed, but only at high values of sound intensity. These and other results are discussed in the context of psychological studies concerning the perception of speech information.