New Better than Used and Other Concepts for a Class of Life Distributions

It is well-known that the inequalities used in the definition of the New Better than Used (N. B. U.) and the New Better than Used in Expectation (N.B.U.E.) concepts, see BARLOW and PROSCHAN (1965, 1975) become equalities if, and only if, the life length of an organism follows an exponential distribution. It is proved in the present paper that these inequalities also reduce to equalities for the class of life distributions that have the “setting the clock back to zero” property. Simple examples of these distributions include the exponential, the linear hazard exponential and the Gompertz distributions. The General Krane distributions (Krane 1963) belong to this class, as well as a recent model introduced by CHIANG and CONFORTI (1989) of a survival distribution in which the hazard rate is a function of the accumulated effect of an individual's continuous exposure to the toxic material in the environment and his biological reaction to the toxin absorbed. As a simple application of the result proved in the paper, the life expectancy of an organism at age γ₀ involved in the N.B.U.E. concept is evaluated for the Gompertzian growth process and for the Chiang and Conforti model.     

On the incompatibility of gompertz or weibull survival dynamics with exponentially distributed individual lifespans

It is well documented, in the biological literature, that many species throughout the animal kingdom exhibit Gompertzian or Weibull-like population level total survival distributions. Many researchers have long assumed, believed, or otherwise postulated that an individual organism, in such a population, survived according to an exponential survival distribution. Using well-known results from reliability theory, it is shown that if every individual in the population has an exponentially distributed lifespan, then a Gompertzian or Weibull-like group/population level dynamics (or any other dynamics with a strictly increasing mortality rate for some interval) is not possible. This implies that, for species with a population level Gompertzian or Weibull (with the mortality rate strictly increasing) survival curve, some or all of the individual organisms must have non-exponentially distributed lifespans.     

The size distribution of human hematogenous metastases.

The development of primary cancers and their subsequent metastases occur through a complex sequence of discrete steps. A hypothesis is proposed here whereby the time available for the growth of metastases is normally distributed, presumably as a consequence of the summation of multiple independently distributed time intervals from each of the steps and of the Central Limit Theorem. For exponentially growing metastases, the corresponding size distribution would be lognormal; Gompertzian growth would imply a modified (Gompertz-normal) distribution, where larger metastases would occur less frequently as a consequence of a decreased growth rate. These two size distributions were evaluated against 18 human autopsy cases where precise size measurements had been collected from over 3900 macroscopic hematogenous organ metastases. The lognormal distribution provided an approximate agreement. Its main deficiency was a tendency to over-represent metastases greater than 10 mm diameter. The Gompertz-normal distribution provided more stringent agreement, correcting for this over-representation. These observations supported the hypothesis of normally distributed growth times, and qualified the utility of the lognormal and Gompertz-normal distributions for the size distribution of metastases.     

Persistence of Repeated Sequences That Evolve by Replication Slippage

The evolution of short repeated sequences by replication slippage under the assumption of selective neutrality is modeled using a linear birth and death process. The equilibrium distribution, the distribution of the life expectancy of a repeated sequence when the process starts from two repeats, the age distribution of repeats, the probability of obtaining two genes with i and j copies which diverged t generations ago and the conditional variance of copy number given the repeat number is more than one are computed. The distributions of life expectancy and age are shown to have long tails. Also the statistic which estimates the conditional variance is shown to have a large coefficient of variation. Using these theoretical results, we develop an approximate test of our model and analyze persistent repeated sequences found in the primate beta-globin gene region and Oenothera chloroplast DNA which are polymorphic within species. We found one sequence in Oenothera chloroplast DNA which does not fit to our neutral model.     

On time-space of nonlinear phenomena with Gompertzian dynamics

This paper describes a universal relationship between time and space for a nonlinear process with Gompertzian dynamics, such as growth. Gompertzian dynamics implicates a coupling between time and space. Those two categories are related to each other through a linear function of their logarithms. Moreover, we demonstrate that the spatial fractal dimension is a function of both scalar time and the temporal fractal dimension. The Gompertz function reflects the equilibrium of regular states, that is, states with dynamics that are predictable for any time-point (e.g., sinusoidal glycolytic oscillations) and chaotic states, that is, states with dynamics that are unpredictable in time, but are characterized by certain regularities (e.g., the existence of strange attractor for any biochemical reaction). We conclude that both this equilibrium and volume of the available complementary Euclidean space determine temporal and spatial expansion of a process with Gompertzian dynamics.     

Health,income, and the preston curve: A long view

Well-being is increasingly viewed as a multidimensional phenomenon, of which income is only one facet. In this paper I focus on another one, health, and look at its synthetic measure, life expectancy at birth, and its relationship with per capita income. International trends of life expectancy and per capita GDP differed during the past 150 years. Life expectancy gains depended on economic growth but also on the advancement in medical knowledge. The pace and breadth of the health transitions drove life expectancy aggregate tendencies and distribution. The new results confirm the relationship between life expectancy and per capita income and its outward shift over time as put forward by Samuel Preston. However, the association between nonlinearly transformed life expectancy and the log of per capita income does not flatten out over time, but becomes convex suggesting more than proportional increases in life expectancy at higher per capita income levels.     

Stochastic dynamics of cancer initiation

Most human cancer types result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Once the first change (or changes) have arisen, tumorigenesis is initiated and the subsequent emergence of additional alterations drives progression to more aggressive and ultimately invasive phenotypes. Elucidation of the dynamics of cancer initiation is of importance for an understanding of tumor evolution and cancer incidence data. In this paper, we develop a novel mathematical framework to study the processes of cancer initiation. Cells at risk of accumulating oncogenic mutations are organized into small compartments of cells and proliferate according to a stochastic process. During each cell division, an (epi)genetic alteration may arise which leads to a random fitness change, drawn from a probability distribution. Cancer is initiated when a cell gains a fitness sufficiently high to escape from the homeostatic mechanisms of the cell compartment. To investigate cancer initiation during a human lifetime, a 'race' between this fitness process and the aging process of the patient is considered; the latter is modeled as a second stochastic Markov process in an aging dimension. This model allows us to investigate the dynamics of cancer initiation and its dependence on the mutational fitness distribution. Our framework also provides a methodology to assess the effects of different life expectancy distributions on lifetime cancer incidence. We apply this methodology to colorectal tumorigenesis while considering life expectancy data of the US population to inform the dynamics of the aging process. We study how the probability of cancer initiation prior to death, the time until cancer initiation, and the mutational profile of the cancer-initiating cell depends on the shape of the mutational fitness distribution and life expectancy of the population.     

In search of the "hair cycle clock": a guided tour

The hair follicle, a unique characteristic of mammals, represents a stem cell-rich, prototypic neuroectodermal-mesodermal interaction system. This factory for pigmented epithelial fibers is unique in that it is the only organ in the mammalian body which, for its entire lifetime, undergoes cyclic transformations from stages of rapid growth (anagen) to apoptosis-driven regression (catagen) and back to anagen, via an interspersed period of relative quiescence (telogen). While it is undisputed that the biological "clock" that drives hair follicle cycling resides in the hair follicle itself, the molecular nature of the underlying oscillator system remains to be clarified. To meet this challenge is of profound general interest, since numerous key problems of modern biology can be studied exemplarily in this versatile model system. It is also clinically important, since the vast majority of patients with hair growth disorders suffers from an undesired alteration of hair follicle cycling. Here, we sketch basic background information and key concepts that one needs to keep in mind when exploring the enigmatic "hair cycle clock"(HCC), and summarize competing models of the HCC. We invite the reader on a very subjective guided tour, which focuses on our own trials, errors, and findings toward the distant goal of unravelling one of the most fascinating mysteries of biology: Why does the hair follicle cycle at all? How does it do it? What are the key players in the underlying molecular controls? Attempting to offer at least some meaningful answers, we share our prejudices and perspectives, and define crucial open questions.     

A principle of fractal-stochastic dualism and Gompertzian dynamics of growth and self-organization

The emergence of Gompertzian dynamics at the macroscopic, tissue level during growth and self-organization is determined by the existence of fractal-stochastic dualism at the microscopic level of supramolecular, cellular system. On one hand, Gompertzian dynamics results from the complex coupling of at least two antagonistic, stochastic processes at the molecular cellular level. It is shown that the Gompertz function is a probability function, its derivative is a probability density function, and the Gompertzian distribution of probability is of non-Gaussian type. On the other hand, the Gompertz function is a contraction mapping and defines fractal dynamics in time-space; a prerequisite condition for the coupling of processes. Furthermore, the Gompertz function is a solution of the operator differential equation with the Morse-like anharmonic potential. This relationship indicates that distribution of intrasystemic forces is both non-linear and asymmetric. The anharmonic potential is a measure of the intrasystemic interactions. It attains a point of the minimum (U(0), t(0)) along with a change of both complexity and connectivity during growth and self-organization. It can also be modified by certain factors, such as retinoids.     

Ambiguity of the solution of the inverse problem of electroencephalography

It has been shown for homogeneous conducting medium that if different stationary distributions of current sources form identical potential fields on the plane boundary, the difference between these two distributions is the distribution of current sources with all multiple moments equaling zero. Examples of current source distributions are presented which satisfy this condition.     

Differentiation and frequency distributions of body weights in plants and animals

The basic and simplest system that one can consider in ecology is a group of individuals of equal age and representing one species, that is, a cohort. This paper is an attempt to show that analysis of such a system may be of great importance to understanding basic ecological problems, such as, intraspecific competition and the dynamics of a single population. It is easy to observe that in even-aged populations individuals differ in weights. A close look can show that weight distributions in even-aged populations may have different skewness. Most common are distributions with coefficients of skewness greater than zero. Sometimes weight distributions are symmetrical or with skewness coefficients less than zero. In a cohort of growing individuals the coefficient of skewness changes with time: most often starting from zero (symmetrical distribution), it increases in time; sometimes after an initial increase it can decrease in the final stage of growth, which is related to an increased mortality of individuals. The rate of change in skewness, and the skewness itself depend on the density of individuals in a cohort and on food conditions. They are greater at higher densities and increase with deteriorating food conditions. Weight distributions are symmetrical at low densities and optimal food conditions. The differences in individual weights measured by variance of weight distributions or coefficient of variation follow the same pattern, but observed changes with time, density and food conditions are not so clear. These conclusions rest upon the review of numerous papers concerning both plants and animals, which is presented in this paper. In the past, the properties of weight distributions in even-aged populations were explained not by interactions between individuals, but rather as a natural outcome of the growth process of non-interacting individuals. The exponential equation of growth, with relative growth rate having a normal distribution in populations, was used to support this hypothesis. Obtained weight distributions were of positive skewness; however, this model, which in fact is able to describe the growth process only in its initial stage, cannot explain the changes of skewness of weight distributions with density and food conditions. A model has been developed which includes competitive interactions among members of even-aged populations to explain observed properties of weight distributions in them. The basic assumption is that intraspecific competition leads to uneven partitioning of resources, which are the object of competition. Functions describing resource partitioning among individuals are included into the model.(ABSTRACT TRUNCATED AT 400 WORDS)     

On the Gompertzian growth in the fractal space-time

An analytical approach to determination of time-dependent temporal fractal dimension b(t)(t) and scaling factor a(t)(t) for the Gompertzian growth in the fractal space-time is presented. The derived formulae take into account the proper boundary conditions and permit a calculation of the mean values b(t)(t) and a(t)(t) at any period of time. The formulae derived have been tested on experimental data obtained by Schrek for the Brown-Pearce rabbit's tumor growth. The results obtained confirm a possibility of successful mapping of the experimental Gompertz curve onto the fractal power-law scaling function y(t)=a(t)tb(t) and support a thesis that Gompertzian growth is a self-similar and allometric process of a holistic nature.     

Comments on "Seed germination as a thermobiological problem"

In the above mentioned article [1] the notion of a time-dependent Gibbs free energy has been introduced to explain the observed time-pattern of embryo growth in seeds. Furthermore, the notion of a 'non-random thermal communication' has been inferred from an inspection of the shapes of germination time distributions. It can be shown, however, that the reasoning leading to the time-dependent thermodynamic potential is based on inappropriate interpretation of kinetic equations, and that the shape of the distributions of germination time might be a natural consequence of the initial distribution of embryo size in the seeds.     

Bounds on life expectancy for the Rayleigh and Weibull distributions

Bounds are presented for the life expectancy or the mean residual life of an individual whose lifetime is a random variable X following a Rayleigh distribution or more generally a Weibull distribution. Simple transformations of the variables give inequalities on the Mills' ratio and the incomplete gamma functions. Some numerical computations are also reported to compare the lower and upper bounds with the exact value of the life expectancy function for several values of the parameter. When the lifetime follows a Gompertz distribution, the problem becomes complicated, and it has not been possible to construct bounds on the life expectancy function. The importance of the Gompertz distribution in the dynamics of normal and tumor growth and in the embryonic and postnatal growth of birds and mammals is demonstrated, and life expectancy is evaluated by numerical methods for a number of parameter values.     

Additive Uncorrelated Relaxed Clock Models for the Dating of Genomic Epidemiology Phylogenies

Phylogenetic dating is one of the most powerful and commonly used methods of drawing epidemiological interpretations from pathogen genomic data. Building such trees requires considering a molecular clock model which represents the rate at which substitutions accumulate on genomes. When the molecular clock rate is constant throughout the tree then the clock is said to be strict, but this is often not an acceptable assumption. Alternatively, relaxed clock models consider variations in the clock rate, often based on a distribution of rates for each branch. However, we show here that the distributions of rates across branches in commonly used relaxed clock models are incompatible with the biological expectation that the sum of the numbers of substitutions on two neighboring branches should be distributed as the substitution number on a single branch of equivalent length. We call this expectation the additivity property. We further show how assumptions of commonly used relaxed clock models can lead to estimates of evolutionary rates and dates with low precision and biased confidence intervals. We therefore propose a new additive relaxed clock model where the additivity property is satisfied. We illustrate the use of our new additive relaxed clock model on a range of simulated and real data sets, and we show that using this new model leads to more accurate estimates of mean evolutionary rates and ancestral dates.     

Subject-specific modeling of the scapula bone tissue adaptation

Adaptation of the scapula bone tissue to mechanical loading is simulated in the current study using a subject-specific three-dimensional finite element model of an intact cadaveric scapula. The loads experienced by the scapula during different types of movements are determined using a subject-specific large-scale musculoskeletal model of the shoulder joint. The obtained density distributions are compared with the CT-measured density distribution of the same scapula. Furthermore, it is assumed that the CT-measured density distribution can be estimated as a weighted linear combination of the density distributions calculated for different loads experienced during daily life. An optimization algorithm is used to determine the weighting factors of fourteen different loads such that the difference between the weighted linear combination of the calculated density distributions and the CT-measured density is minimal. It is shown that the weighted linear combination of the calculated densities matches the CT-measured density distribution better than every one of the density distributions calculated for individual movements. The weighting factors of nine out of fourteen loads were estimated to be zero or very close to zero. The five loads that had larger weighting factors were associated with either one of the following categories: (1) small-load small-angle abduction or flexion movements that occur frequently during our daily lives or (2) large-load large-angle abduction or flexion movements that occur infrequently during our daily lives.     

Graduated change of life expectancy in mice in ontogenesis

Life expectancy of descendants of a normal female mouse and a male with an inherited growth inhibition mutation discovered in a laboratory population was investigated. The hereditability of the characteristic allows us to consider it a result of mutation. It was shown that, in mice, the curve of dependence of life expectancy on their serial number in a row of increase in life expectancy (curve of rank distribution) has step-like shape for mutant males and females, as well as for males with normal development. The first grade of mice death on the curve of rank distribution was observed at one month after their birth and was characteristic only of males and females with a mutation during the period of maximum lag in weight as compared with their normal relatives. The surviving mutants catch up to the normally developing individuals within two months and externally become indistinguishable from them. The subsequent grades of death in mutants and normal males coincide on the time axis. The steps are absent on the rank curves of life expectancy of normally developing females. The time intervals between the steps are reproduced in parallel groups of mice and, hence, are not casual deviations from theoretical curves and are of a regular nature. The discovered phenomenon is interpreted within the scope of a hypothesis about the realization of the genetic program of ontogenesis, which provides periodic change of vitality stages with stages of sensitivity to external risk factors, which increase the probability of death, by mice. Absence of such stages in the group of normally developing females can be explained by shifts in development, which are produced by the irregular performance of reproductive functions.     

Mechanisms of aging: an appraisal of the oxidative stress hypothesis

The main purpose of this article is to provide a critical overview of the currently available evidence bearing on the validity of the oxidative stress hypothesis of aging, which postulates that senescence-associated attenuations in physiological functions are caused by molecular oxidative damage. Several lines of correlative evidence support the predictions of the hypothesis, e.g., macromolecular oxidative damage increases with age and tends to be associated with life expectancy of organisms. Nevertheless, a direct link between oxidative stress and aging has not as yet been established. Single gene mutations have been reported to extend the life spans of lower organisms, such as nematodes and insects; however, such prolongations of chronological clock time survival are usually associated with decreases in the rate of metabolism and reproductive output without affecting the metabolic potential, i.e., the total amount of energy consumed during life. Studies on genetic manipulations of the aging process have often been conducted on relatively short-lived strains that are physiologically weak, whereby life-span extensions can not be unambiguously assigned to a slowing effect on the rate of aging. It is concluded that although there is considerable evidence implicating oxidative stress in the aging process, additional evidence is needed to clearly define the nature of the involvement.     

Four easy pieces: mechanisms underlying circadian regulation of growth and development

The circadian clock confers rhythms of approximately 24 hours to biological events. It elevates plant fitness by allowing plants to anticipate predictable environmental changes and organize life process to coincide with the most favorable environmental conditions. Many developmental events are circadian regulated to ensure that growth occurs at the ideal time or season relative to available resources. Circadian clock control over growth and development is often achieved through regulation of key phytohormone action. Circadian influence over the genome is widespread and the clock modulates genes involved in phytohormone synthesis and signaling, in addition to other pathways shaping growth and development. This review presents four nonmutually exclusive mechanisms by which temporal information is gleaned from the core oscillator and passed to pathways regulating plant growth and development.     

DSP 4, a neurotoxic agent that destroys noradrenergic endings, selectively increases "expectancy" in the cat

Cats were treated with DSP 4, a neurotoxic agent known to destroy central noradrenergic endings. A significant increase was subsequently noticed in the amount of time spent by the treated animals in an attitude of "expectancy", i.e. of motionless waiting for an "event to occur". They even developed this attitude when no such real situation existed. Concomitantly, an increase was noticed in the power of the 14 Hz electrocortical rhythms recorded over the somatic sensory cortex. These patterns, designated as "mu" rhythms, had previously been shown to characterize this particular type of attentive state. The present data tends to confirm our previous hypothesis, that immobile expectancy and its accompanying electrocortical pattern are under a noradrenergic inhibitory control.