The age-related dynamics of mortality and the Gompertz-Makeham law

Using the statistics of mortality of Caucasian population of 48 states of the USA (1969-1971) it was demonstrated that the real age dynamics of human mortality may differ significantly both from the Gompertz law and from the Gompertz-Makeham law. Using of the Gompertz-Makeham formula leads to appearance of negative A value in 77 cases out of 96. This makes it difficult to interpret this parameter as a "background" component of mortality. Using of the Gompertz formula in different age groups leads uncoordinated changes in alpha and R0 values in every state. Hence, it is impossible to plot geographically stable characters for Gompertz parameters alpha for subsequent epidemiological analysis. The "aging rate", estimated by parameter is not stable throughout the life span of 30-92 years, but changes with certain pattern.     

Dietary restriction of rodents decreases aging rate without affecting initial mortality rate – a meta‐analysis

Dietary restriction (DR) extends lifespan in multiple species from various taxa. This effect can arise via two distinct but not mutually exclusive ways: a change in aging rate and/or vulnerability to the aging process (i.e. initial mortality rate). When DR affects vulnerability, this lowers mortality instantly, whereas a change in aging rate will gradually lower mortality risk over time. Unraveling how DR extends lifespan is of interest because it may guide toward understanding the mechanism(s) mediating lifespan extension and also has practical implications for the application of DR. We reanalyzed published survival data from 82 pairs of survival curves from DR experiments in rats and mice by fitting Gompertz and also Gompertz–Makeham models. The addition of the Makeham parameter has been reported to improve the estimation of Gompertz parameters. Both models separate initial mortality rate (vulnerability) from an age‐dependent increase in mortality (aging rate). We subjected the obtained Gompertz parameters to a meta‐analysis. We find that DR reduced aging rate without affecting vulnerability. The latter contrasts with the conclusion of a recent analysis of a largely overlapping data set, and we show how the earlier finding is due to a statistical artifact. Our analysis indicates that the biology underlying the life‐extending effect of DR in rodents likely involves attenuated accumulation of damage, which contrasts with the acute effect of DR on mortality reported for Drosophila. Moreover, our findings show that the often‐reported correlation between aging rate and vulnerability does not constrain changing aging rate without affecting vulnerability simultaneously.     

The reliability theory of aging and longevity.

Reliability theory is a general theory about systems failure. It allows researchers to predict the age-related failure kinetics for a system of given architecture (reliability structure) and given reliability of its components. Reliability theory predicts that even those systems that are entirely composed of non-aging elements (with a constant failure rate) will nevertheless deteriorate (fail more often) with age, if these systems are redundant in irreplaceable elements. Aging, therefore, is a direct consequence of systems redundancy. Reliability theory also predicts the late-life mortality deceleration with subsequent leveling-off, as well as the late-life mortality plateaus, as an inevitable consequence of redundancy exhaustion at extreme old ages. The theory explains why mortality rates increase exponentially with age (the Gompertz law) in many species, by taking into account the initial flaws (defects) in newly formed systems. It also explains why organisms "prefer" to die according to the Gompertz law, while technical devices usually fail according to the Weibull (power) law. Theoretical conditions are specified when organisms die according to the Weibull law: organisms should be relatively free of initial flaws and defects. The theory makes it possible to find a general failure law applicable to all adult and extreme old ages, where the Gompertz and the Weibull laws are just special cases of this more general failure law. The theory explains why relative differences in mortality rates of compared populations (within a given species) vanish with age, and mortality convergence is observed due to the exhaustion of initial differences in redundancy levels. Overall, reliability theory has an amazing predictive and explanatory power with a few, very general and realistic assumptions. Therefore, reliability theory seems to be a promising approach for developing a comprehensive theory of aging and longevity integrating mathematical methods with specific biological knowledge.     

Life equations for the senescence process

The Gompertz law of mortality quantitatively describes the mortality rate of humans and almost all multicellular animals. However, its underlying kinetic mechanism is unclear. The Gompertz law cannot explain the mortality plateau at advanced ages and cannot give an explicit relationship between temperature and mortality. In this study a reaction kinetics model with a time dependent rate coefficient is proposed to describe the survival and senescence processes. A temperature-dependent mortality function was derived. The new mortality function becomes the Gompertz mortality function with the same relationship of parameters prescribed by the Strehler–Mildvan correlation when age is smaller than a characteristic value δ, and reaches the mortality plateau when age is greater than δ. A closed-form analytical expression for describing the relationship of average lifespan with temperature and other equations are derived from the new mortality function. The derived equations can be used to estimate the limit of average lifespan, predict the maximal longevity, calculate the temperature coefficient of lifespan, and explain the tendency of the survival curve. This prediction is consistent with the most recently reported mortality trajectories for single-year birth cohorts. This study suggests that the senescence process results from the imbalance between damaging energy and protecting energy for the critical chemical substance in the body. The rate of senescence of the organism increases while the protecting energy decreases. The mortality plateau is reached when the protecting energy decreases to its minimal levels. The decreasing rate of the protecting energy is temperature-dependent. This study is exploring the connection between the biochemical mechanism and demography.     

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.     

Within-population variation in cytoplasmic genes affects female life span and aging in Drosophila melanogaster

It has been suggested that mitochondrial DNA (mtDNA) may play an important role in aging. Yet, few empirical studies have tested this hypothesis, partly because the degree of sequence polymorphism in mtDNA is assumed to be low. However, low sequence variation may not necessarily translate into low phenotypic variation. Here, we report an experiment that tests whether there is within-population variation in cytoplasmic genes for female longevity and senescence. To achieve this, we randomly selected 25 "mitochondrial founders" from a single, panmictic population of Drosophila melanogaster and used these founders to generate distinct "mt" lines in which we controlled for the nuclear background by successive backcrossing. Potential confounding effects of cytoplasmically transmitted bacteria were eliminated by tetracycline treatment. The mt lines were then assayed for differences in longevity, Gompertz intercept (frailty), and demographic rate of change in mortality with age (rate-of-senescence) in females. We found significant cytoplasmic effects on all three variables. This provides evidence that genetic variation in cytoplasmic genes, presumably mtDNA, contributes to variation in female mortality and aging.     

A simple derivation of the Gompertz law for human mortality

The Gompertz law of dependence of human mortality rate on age is derived from a simple model of death as a result of an exponentially rare escape of abnormal cells from immunological response.     

Beyond the proportional frailty model: Bayesian estimation of individual heterogeneity on mortality parameters

Today, we know that demographic rates can be greatly influenced by differences among individuals in their capacity to survive and reproduce. These intrinsic differences, commonly known as individual heterogeneity, can rarely be measured and are thus treated as latent variables when modeling mortality. Finite mixture models and mixed effects models have been proposed as alternative approaches for inference on individual heterogeneity in mortality. However, in general models assume that individual heterogeneity influences mortality proportionally, which limits the possibility to test hypotheses on the effect of individual heterogeneity on other aspects of mortality such as ageing rates. Here, we propose a Bayesian model that builds upon the mixture models previously developed, but that facilitates making inferences on the effect of individual heterogeneity on mortality parameters other than the baseline mortality. As an illustration, we apply this framework to the Gompertz–Makeham mortality model, commonly used in human and wildlife studies, by assuming that the Gompertz rate parameter is affected by individual heterogeneity. We provide results of a simulation study where we show that the model appropriately retrieves the parameters used for simulation, even for low variances in the heterogeneous parameter. We then apply the model to a dataset on captive chimpanzees and on a cohort life table of 1751 Swedish men, and show how model selection against a null model (i.e., without heterogeneity) can be carried out.     

The influence of genes on the aging process of mice: a statistical assessment of the genetics of aging

Genetic interventions that accelerate or retard aging in mice are crucial in advancing our knowledge over mammalian aging. Yet determining if a given intervention affects the aging process is not straightforward since, for instance, many disease-causing mutations may decrease life span without affecting aging. In this work, we employed the Gompertz model to determine whether several published interventions previously claimed to affect aging in mice do indeed alter the aging process. First, we constructed age-specific mortality tables for a number of mouse cohorts used in longevity experiments and calculated the rate at which mortality increases with age. Estimates of age-independent mortality were also calculated. We found no statistical evidence that GHRHR, IGF1R, INSR, PROP1, or TRX delay or that ATM + TERC, BubR1, klotho, LMNA, PRDX1, p53, WRN + TERC, or TOP3B accelerate mouse aging. Often, changes in the expression of these genes affected age-independent mortality and so they may prove useful to other aspects of medicine. We found statistical evidence that C/EBP, MSRA, SHC1, growth hormone, GHR, PIT1, and PolgA may influence aging in mice. These results were interpreted together with age-related physiological and pathological changes and provide novel insights regarding the role of several genes in the mammalian aging process.     

The adequacy of aging techniques in vertebrates for rapid estimation of population mortality rates from age distributions

As a key parameter in population dynamics, mortality rates are frequently estimated using mark–recapture data, which requires extensive, long‐term data sets. As a potential rapid alternative, we can measure variables correlated to age, allowing the compilation of population age distributions, from which mortality rates can be derived. However, most studies employing such techniques have ignored their inherent inaccuracy and have thereby failed to provide reliable mortality estimates. In this study, we present a general statistical model linking birth rate, mortality rate, and population age distributions. We next assessed the reliability and data needs (i.e., sample size) for estimating mortality rate of eight different aging techniques. The results revealed that for half of the aging techniques, correlations with age varied considerably, translating into highly variable accuracies when used to estimate mortality rate from age distributions. Telomere length is generally not sufficiently correlated to age to provide reliable mortality rate estimates. DNA methylation, signal‐joint T‐cell recombination excision circle (sjTREC), and racemization are generally more promising techniques to ultimately estimate mortality rate, if a sufficiently high sample size is available. Otolith ring counts, otolithometry, and age‐length keys in fish, and skeletochronology in reptiles, mammals, and amphibians, outperformed all other aging techniques and generated relatively accurate mortality rate estimation with a sample size that can be feasibly obtained. Provided the method chosen is minimizing and estimating the error in age estimation, it is possible to accurately estimate mortality rates from age distributions. The method therewith has the potential to estimate a critical, population dynamic parameter to inform conservation efforts within a limited time frame as opposed to mark–recapture analyses.     

An explanation for negligible senescence in animals

Negligible or negative senescence occurs when mortality risk is stable or decreases with age, and has been observed in some wild animals. Age‐independent mortality in animals may lead to an abnormally long maximum individual lifespans and be incompatible with evolutionary theories of senescence. The reason why there is no evidence of senescence in these animals has not been fully understood. Recovery rates are usually very low for wild animals with high dispersal ability and/or small body size (e.g., bats, rodents, and most birds). The only information concerning senescence for most of these species is the reported lifespan when individuals are last seen or caught. We deduced the probability density function of the reported lifespan based on the assumption that the real lifespan corresponding to Weibull or Gompertz distribution. We show that the magnitude of the increase in mortality risk is largely underestimated based on the reported lifespans with low recovery probability. The risk of mortality can aberrantly appear to have a negative correlation with age when it actually increases with increasing lifespan. We demonstrated that the underestimated aging rate for wild animals with low recovery probability can be generalizable to any aging models. Our work provides an explanation for the appearance of negligible senescence in many wild animals. Humans attempt to obtain insights from other creatures to better understand our own biology and its gain insight into how to enhance and extended human health. Our advice is to take a second glance before admiring the negligible senescence in other animals. This ability to escape from senescence is possibly only as beautiful illusion in animals.     

Critique of the demographic evidence for 'late-life non-senescence'

Although the Gompertz formula accurately describes observed mortality distributions over most of their extent, their 'tail' is much longer than that of a Gompertz curve fitted to the whole data set. A simple candidate explanation is that the longest-lived subset of any population will necessarily be enriched in individuals that age more slowly than the average of that population. However, some investigators have suggested that, instead, individuals actually cease to senesce after a certain age. Here, using a new approach to determining the best-fit degree of heterogeneity in the Gompertz slope parameter, it is shown that observed distributions can in fact be fit quite accurately by purely 'heterogeneous Gompertz' curves. Either explanation may therefore be correct.     

Age-specific mortality rates of reproducing and non-reproducing males of Drosophila melanogaster

The proximate and evolutionary causes of the levelling of mortality rates at late ages, observed in several species, remain obscure. To investigate the causes of mortality levelling late in life in Drosophila melanogaster, we examined the effect of reproduction on mortality patterns, by conducting population cage experiments with a total of more than 45,000 individuals. Several different genotypes of reproducing and non-reproducing males from F(1) crosses among isogenic lines were studied. Our results suggest that significant mortality levelling can occur even in non-reproducing males, but that reproduction also significantly affects mortality patterns. The results show that mortality levelling is strongly affected by the Gompertz initial mortality rate and exponential rate of increase parameters, probably through the effects of heterogeneity in mortality risks.     

Toward reconciling inferences concerning genetic variation in senescence in Drosophila melanogaster.

Standard models for senescence predict an increase in the additive genetic variance for log mortality rate late in the life cycle. Variance component analysis of age-specific mortality rates of related cohorts is problematic. The actual mortality rates are not observable and can be estimated only crudely at early ages when few individuals are dying and at late ages when most are dead. Therefore, standard quantitative genetic analysis techniques cannot be applied with confidence. We present a novel and rigorous analysis that treats the mortality rates as missing data following two different parametric senescence models. Two recent studies of Drosophila melanogaster, the original analyses of which reached different conclusions, are reanalyzed here. The two-parameter Gompertz model assumes that mortality rates increase exponentially with age. A related but more complex three-parameter logistic model allows for subsequent leveling off in mortality rates at late ages. We find that while additive variance for mortality rates increases for late ages under the Gompertz model, it declines under the logistic model. The results from the two studies are similar, with differences attributable to differences between the experiments.     

Mortality shifts in Caenorhabditis elegans: remembrance of conditions past

The analysis of age‐specific mortality can yield insights into how anti‐aging interventions operate that cannot be matched by simple assessment of longevity. Mortality, as opposed to longevity, can be used to assess the effects of an anti‐aging intervention on a daily basis, rather than only after most animals have died. Various gerontogene mutations in Caenorhabditis elegans have been shown to increase longevity as much as tenfold and to decrease mortality at some ages even more. Environmental alterations, such as reduced food intake (dietary restriction) and lower temperature also result in reduced mortality soon after the intervention. Here, we ask how soon anti‐aging interventions, applied during adult life, affect age‐specific mortality in nematodes. Using maximum likelihood analysis, we estimated the Gompertz parameters after shifts of temperature, and of food concentration and maintenance conditions. In separate experiments, we altered expression of age‐1 and daf‐16, using RNAi. Using about 44 000 nematodes in total, to examine daily mortality, we find that for both types of environmental shift, mortality responded immediately in the first assessment, while RNAi‐induced changes resulted in a slower response, perhaps due to delayed mechanics of RNAi action. However, under all conditions there is a permanent ‘memory’ of past states, such that the initial mortality component [a] of the Gompertz equation [μ(x) = ae^bx] bears a permanent ‘imprint’ of that earlier state. However, ‘b’ (the rate of mortality increase with age) is always specified by the current conditions.     

Evolution of the rate of biological aging using a phenotype based computational model

In this work I introduce a simple model to study how natural selection acts upon aging, which focuses on the viability of each individual. It is able to reproduce the Gompertz law of mortality and can make predictions about the relation between the level of mutation rates (beneficial/deleterious/neutral), age at reproductive maturity and the degree of biological aging. With no mutations, a population with low age at reproductive maturity R stabilizes at higher density values, while with mutations it reaches its maximum density, because even for large pre-reproductive periods each individual evolves to survive to maturity. Species with very short pre-reproductive periods can only tolerate a small number of detrimental mutations. The probabilities of detrimental (P_d) or beneficial (P_b) mutations are demonstrated to greatly affect the process. High absolute values produce peaks in the viability of the population over time. Mutations combined with low selection pressure move the system towards weaker phenotypes. For low values in the ratio P_d/P_b, the speed at which aging occurs is almost independent of R, while higher values favor significantly species with high R. The value of R is critical to whether the population survives or dies out. The aging rate is controlled by P_d and P_b and the amount of the viability of each individual is modified, with neutral mutations allowing the system more “room” to evolve. The process of aging in this simple model is revealed to be fairly complex, yielding a rich variety of results.