Mathematical modeling of the aging processes and the mechanisms of mortality: paramount role of heterogeneity

Time-evolution of age-dependent mortality patterns in mathematical model of heterogeneous human population

The widely-known Gompertz law of mortality states the exponential increase of mortality with age in human populations. Such an exponential increase is observed at the adulthood span, roughly after the reproductive period, while mortality data at young and extremely old ages deviate from it. The heterogeneity of human populations, i.e. the existence of subpopulations with different mortality dynamics, is a useful consideration that can explain age-dependent mortality patterns across the whole life-course. A simple mathematical model combining the heterogeneity of populations with an assumption that the mortality in each subpopulation grows exponentially with age has been proven to be capable of reproducing the entire mortality pattern in a human population including the observed peculiarities at early- and late-life intervals. In this work we fit this model to actual (Swedish) mortality data for consecutive periods and consequently describe the evolution of mortality dynamics in terms of the evolution of the model parameters over time. We have found that the evolution of the model parameters validates the applicability of the compensation law of mortality to each subpopulation separately. Furthermore, our study has indicated that the population structure changes so that the population tends to become more homogeneous over time. Finally, our analysis of the decrease of the overall mortality in a population over time has shown that this decrease is mainly due to a change in the population structure and to a lesser extent to a reduction of mortality in each of the subpopulations, the latter being represented by an alteration of the parameters that outline the exponential dynamics.

Lifespan and aggregate size variables in specifications of mortality or survivorship

A specification of mortality or survivorship provides respective explicit details about mortality's or survivorship's relationships with one or more other variables (e.g., age, sex, etc.). Previous studies have discovered and analyzed diverse specifications of mortality or survivorship; these discoveries and analyses suggest that additional specifications of mortality or survivorship have yet to be discovered and analyzed. In consistency with previous research, multivariable limited powered polynomials regression analyses of mortality and survivorship of selected humans (Swedes, 1760-2008) and selected insects (caged medflies) show age-specific, historical-time-specific, environmental-context-specific, and sex-specific mortality and survivorship. These analyses also present discoveries of hitherto unknown lifespan-specific, contemporary-aggregate-size-specific, and lifespan-aggregate-size-specific mortality and survivorship. The results of this investigation and results of previous research help identify variables for inclusion in regression models of mortality or survivorship. Moreover, these results and results of previous research strengthen the suggestion that additional specifications of mortality or survivorship have yet to be discovered and analyzed, and they also suggest that specifications of mortality and survivorship indicate corresponding specifications of frailty and vitality. Furthermore, the present analyses reveal the usefulness of a multivariable limited powered polynomials regression model-building approach. This article shows that much has yet to be learned about specifications of mortality or survivorship of diverse kinds of individuals in diverse times and places.

Possible role of NF-κB in hormesis during ageing

Hormesis in ageing is probably represented by mild stress-induced stimulation of protective mechanisms in cells and organisms resulting in biologically beneficial effects. Mild stress and hormetins may act on bifurcation points in the complex network of cell signaling and transcription factors, often turning homeodynamics to health or survival. Several signaling pathways activated by diverse stimuli and by stress response converge on NF-κB activation, resulting in a regulatory system characterized by high complexity. NF-κB behaves as a chaotic oscillator and it is increasingly recognized that the number of components that impinges upon phenotypic outcomes of signal transduction pathways may be higher than those taken into consideration from canonical pathway representations. NF-κB is closely related to other important upstream signaling networks, creating chaotic oscillators with other receptor-related kinases and targeting hubs for hormesis. The great bulk of natural hormetins acts on these signaling pathways, while NF-κB appears as a key regulatory factor in this context. Due to its tight relationship with main signaling system NF-κB plays a fundamental role in stress response, apoptosis and autophagy and appears to be a possible target for hormesis in ageing.

A program for solving the brain ischemia problem

Our recently described nonlinear dynamical model of cell injury is here applied to the problems of brain ischemia and neuroprotection. We discuss measurement of global brain ischemia injury dynamics by time course analysis. Solutions to proposed experiments are simulated using hypothetical values for the model parameters. The solutions solve the global brain ischemia problem in terms of "master bifurcation diagrams" that show all possible outcomes for arbitrary durations of all lethal cerebral blood flow (CBF) decrements. The global ischemia master bifurcation diagrams: (1) can map to a single focal ischemia insult, and (2) reveal all CBF decrements susceptible to neuroprotection. We simulate measuring a neuroprotectant by time course analysis, which revealed emergent nonlinear effects that set dynamical limits on neuroprotection. Using over-simplified stroke geometry, we calculate a theoretical maximum protection of approximately 50% recovery. We also calculate what is likely to be obtained in practice and obtain 38% recovery; a number close to that often reported in the literature. The hypothetical examples studied here illustrate the use of the nonlinear cell injury model as a fresh avenue of approach that has the potential, not only to solve the brain ischemia problem, but also to advance the technology of neuroprotection.

Aging performance for masters records in athletics, swimming, rowing, cycling, triathlon, and weightlifting

Record performances for Masters sporting events for swimming, cycling, triathlon, rowing, and weightlifting were analyzed and then compared with the authors' previously published results for Masters running, walking, and jumping sports events. Records were normalized using the 30s age records as a baseline, and studied through the various age ranges to the 90s. A curvilinear mathematical model [y = 1 - exp((T - T(0))/τ)] was again used for the major comparisons, along with slope changes using a linear model [y = α(T -T'0)] across the age groupings. All sports declined with increasing age, with rowing showing the least deterioration. Performances in running, swimming, and walking were reasonably well maintained, followed by greater decline with age for cycling, triathlon, and jumping events. Weightlifting showed the fastest and greatest decline with increasing age. The relative performances for women, when compared with men's performances for these Masters events, was approximately 80% to 85%, with jumping at 73% and weightlifting at 52%. These relative performances compared with World Record comparisons of approximately 90% (with weightlifting at approximately 75%). All these results show no greater decline with age for endurance events over the sprint events, though there was a greater decline for the strength events of weightlifting and jumping. There may be real physiological differences for these strength events, or there may be other explanations such as training or competitive considerations or smaller numbers participating.

Computational systems biology of aging

Computational systems biology is expected to make major contributions to unravel the complex molecular mechanisms underlying the progression of aging in cells, tissues, and organisms. The development of computational approaches is, however, challenged by a wide spectrum of aging mechanisms participating on different levels of biological organization. The tight connectivity between the molecular constituents, functions, and cell states requires frameworks and strategies that extend beyond current practice to model, simulate, and predict the progression of aging and the emerging aging phenotypes. We provide a general overview of the specific computational tasks and opportunities in aging research, and discuss some illustrative systems level concepts in more detail. One example provided here is the assembly of a conceptual whole cell model that considers the temporal dynamics of the aging process grounded on molecular mechanisms. Another application is the assembly of interactomes, such as protein networks that allow us to analyze changes in network topology and interaction of proteins that have been implicated in aging with other cellular constituents and processes. We introduce the necessary key steps to build these applications and discuss their merits and future extensions for aging research. WIREs Syst Biol Med 2011 3 414-428 DOI: 10.1002/wsbm.126

Rule-based cell systems model of aging using feedback loop motifs mediated by stress responses

Investigating the complex systems dynamics of the aging process requires integration of a broad range of cellular processes describing damage and functional decline co-existing with adaptive and protective regulatory mechanisms. We evolve an integrated generic cell network to represent the connectivity of key cellular mechanisms structured into positive and negative feedback loop motifs centrally important for aging. The conceptual network is casted into a fuzzy-logic, hybrid-intelligent framework based on interaction rules assembled from a priori knowledge. Based upon a classical homeostatic representation of cellular energy metabolism, we first demonstrate how positive-feedback loops accelerate damage and decline consistent with a vicious cycle. This model is iteratively extended towards an adaptive response model by incorporating protective negative-feedback loop circuits. Time-lapse simulations of the adaptive response model uncover how transcriptional and translational changes, mediated by stress sensors NF-κB and mTOR, counteract accumulating damage and dysfunction by modulating mitochondrial respiration, metabolic fluxes, biosynthesis, and autophagy, crucial for cellular survival. The model allows consideration of lifespan optimization scenarios with respect to fitness criteria using a sensitivity analysis. Our work establishes a novel extendable and scalable computational approach capable to connect tractable molecular mechanisms with cellular network dynamics underlying the emerging aging phenotype.

The global process of aging disturbs a broad range of cellular mechanisms in a complex fashion and is not well understood. One important goal of computational approaches in aging is to develop integrated models in terms of a unifying aging theory, predicting progression of aging phenotypes grounded on molecular mechanisms. However, current experimental data incoherently reflects many isolated processes from a large diversity of approaches, biological model systems, and species, which makes such integration a challenging task. In an attempt to close this gap, we iteratively develop a fuzzy-logic cell systems model considering the interplay of damage, metabolism, and signaling by positive and negative feedback-loop motifs using relationships drawn from literature data. Because cellular biodynamics may be considered a complex control system, this approach seems particularly suitable. Here, we demonstrate that rule-based fuzzy-logic models provide semi-quantitative predictions that enhance our understanding of complex and interlocked molecular mechanisms and their implications on the aging physiome.

The shape of things to come: using models with physiological structure to predict mortality trajectories

If mortality rate is viewed as the outcome of processes of behavior, growth and reproduction, then it should be possible to predict mortality rate as a result of those processes. We provide two examples of how this may be done. In the first, we use the method of linear chains to treat mortality that is the result of multiple physiological processes, some of which may have delays. In the second, we assume that mortality is the result of damage associated with growth and metabolism. Both approaches lead to a rich diversity of predicted mortality trajectories. Although many of these look Gompertzian at young ages, the behavior at older ages depends upon the details of the physiological models.

Use of a mathematical model in the analysis of survival curves of Daphnia magna exposed to toxicants

Chronic toxicity tests carried out on species of the genus Daphnia (Crustacea: Cladocera) are bioassays commonly used in ecotoxicology. Mortality in the cohorts exposed to toxicants can be examined by the analysis of survival curves. The shape of these curves may be very different because of inter-individual heterogeneity: the less rectangular is the shape the more different is the probability of dying of daphnids in the cohort. Aim of this paper is to analyze the characteristics-in particular the shape-of survival curves of cohorts of Daphnia magna exposed to heavy metals in chronic toxicity tests. Experimental curves were fitted by a mathematical model recently proposed, which puts emphasis just on this heterogeneity, described by the parameter of the model S(0). The information contained in S(0) is very useful for the present purposes: the higher the value of S(0) the less rectangular the shape of the curve and, as a consequence, the higher the age-at-death heterogeneity of the cohort is thought to be. The model contains a second parameter, omega, representing the maximum potential ability of the individuals to survive in a specific environment and is related with the maximal life span. The model fitted well the survival curves in most cases and both S(0) and omega showed statistically different values between treatments, useful for comparisons. It was concluded that S(0) provides a quantitative estimation of curve "rectangularization", useful to check different sensitivities to a specific toxicant concentration among daphnids belonging to the same cohort, while omega provides an estimate of maximal life span.

Chronic antigenic stress, immunosenescence and human survivorship over the 3 last centuries: heuristic value of a mathematical model

In previous investigations we pursued the hypothesis that lifelong, chronic antigenic load (CAL) is the major driving force of immunosenescence, which impacts on human lifespan by reducing the number of virgin antigen-non experienced (ANE) T cells, and filling the immunological space with expanded clones of memory and effector, antigen-experienced (AE) T cells. A model has been proposed to relate CAL with the conversion rate from ANE to AE CD8+ T cells. In addition, in order to account for individual variations of immunosenescence and lifespan, a noise term has been introduced to describe the individual fluctuations of CAL. This model was able to follow with a reasonable approximation the age-related decrease of ANE CD8+ T cells, as well as human survival curves. In this paper we extend this approach to historical survival curves, starting from 1750 until present days, and show that the quality of the fit of historical demographic data improves as we approach the recent, quantitative and qualitative decrease of CAL. Indeed, the almost linearity in the increase of lifespan and in the decrease of the noise fluctuation amplitude within this historical period suggests that the improvement of life conditions has steadily lowered the intensity of CAL and restricted the variability which results from the interaction between the individuals and their immunological environment. On the whole, this approach allows to appreciate when and how immunosenescence has started to impact on survivorship, and predicts its increasing, crucial role in explaining human mortality in hygienized, economically developed societies.

Influence of age on circadian rhythms of adrenoceptors in brain cortex, heart and submandibular glands of BALB/c mice: when circadian studies are not only useful but necessary

In the present paper we study circadian rhythm variation of alpha1-adrenoceptors (alpha(1)ARs) and beta-adrenoceptors (betaARs) characteristics during aging. Experiments have been performed on mice's whole brain cortex, heart and submandibular glands, which have physiologically relevant regulatory functions at different ages. We used 36 young (4 months) and 36 old (26 months) mice; six animals from each group were sacrificed every 4h, starting at 8.00 a.m. Both the alpha(1)ARs and betaARs density in heart and brain and the betaARs of submandibular glands are significantly lower in old mice with respect to the young ones. The difference, however, if the comparison is performed only at a particular time, for example 8.00 a.m., is not always significant. The alpha(1)ARs density in submandibular glands does not show significant difference with aging, but opposite phases are observed. This study demonstrates that in dynamical systems such as receptors, single point analysis is not so suitable for a correct estimate of adrenoceptors density variation between young and old mice.