Henry R. Hirsch

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 Bidders hypothesis that aging is a by-product of the cessation of growth.

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.

Determination of the cell doubling-time distribution from culture growth-rate data

Abstract The fundamental integral equation governing the growth of a cell culture is solved for the cell doubling-time distribution with the use of Laplace transform methods. Both analytic and numerical solutions are discussed. The statistical moments of the doubling-time distribution are obtained without inverting the Laplace transform of the distribution. A very simple equation makes it possible to calculate the standard deviation of the doubling-time distribution from the decay of synchronization of a synchronized cell culture.

Decay of cell synchronization: solutions of the cell-growth equation.

Growth-rate functions in analytic form have been obtained for cell cultures in which the doubling times follow the Gaussian and Poisson distributions. The growth-rate functions are calculated by using Laplace transforms to solve an integral equation previously presented. Oscillatory solutions result if a substantial fraction of the cells in a culture are synchronized to divide at some particular time. The synchrony and, hence, the oscillatory character of the growth-rate function eventually disappear because of the non-zero variance of the doubling-time distribution. If their variances are sufficiently small, the Gaussian and Poisson doubling-time distributions lead to growth-rate functions that become identical in the limit of large time.

Dynamics of growth in mammalian diploid tissue cultures

Abstract The growth of a mammalian diploid tissue culture is described by a simple set of differential equations. The equations are motivated by data obtained by Merz and Ross which show that the fraction of sterile cells in a culture increases approximately exponentially with passage number. Solutions of the equations reproduce phases II and III of the culture growth pattern observed by Hayflick. The exact form of the solutions depends on a specific sterility-rate function which is defined by analogy with specific death-rate functions used in theories of aging. An autocatalytic theory leads to the use of an exponential function of time, while a multistep or a target theory leads to the use of a power function. Either of these functions yields solutions of the differential growth equations which are compatible with the observations made by Merz and Ross.

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.

An analog of cell-culture growth

Abstract A formal analogy is drawn between the growth of a cell culture and the amplification process in a regenerative feedback amplifier. Quantitative comparisons between these two systems are given. A relatively simple analog computer is described with which it is possible to study relationships among the growth-rate, initial-state, and doubling-time-distribution functions of the cell culture. The doubling-time distribution function is represented by the impulse response of a filter. The statistical moments of the impulse response can be used in the synthesis of the filter.

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.

The multistep theory of aging: relation to the forbidden-clone theory. A theoretical article

Abstract The multistep theory of aging is presented in a revised form. The pathogenesis of disease is represented by multistep processes consisting of sequences of steps linking states that are organized in serial chains. Cells or other disease-related entities make irreversible transitions from one state in a chain to the next. The clinical manifestations of a given disease commerce when a sufficient number of cells has populated the final state of each member of a set of multistep processes. The mean number of cells in the final state of each process is calculated with the use of Laplace transform techniques. The result is a sum of decaying exponential functions of time which approaches the power function obtained by previous investigators if the probability of transition between states is small. Poisson statistics are used to calculate the probability of onset of disease as a function of the mean numbers of cells in the final states of the multistep processes. Under certain conditions, the multistep theory yields the same expression as the forbidden-clone theory for the incidence of disease as a function of age. Thus age-specific disease-prevalence data which has been advanced to support the forbidden-clone theory is equally supportive of the multistep theory.

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.