A computational approach to studying ageing at the individual level

Abstract

The ageing process is actively regulated throughout an organism's life, but studying the rate of ageing in individuals is difficult with conventional methods. Consequently, ageing studies typically make biological inference based on population mortality rates, which often do not accurately reflect the probabilities of death at the individual level. To study the relationship between individual and population mortality rates, we integrated in vivo switch experiments with in silico stochastic simulations to elucidate how carefully designed experiments allow key aspects of individual ageing to be deduced from group mortality measurements. As our case study, we used the recent report demonstrating that pheromones of the opposite sex decrease lifespan in Drosophila melanogaster by reversibly increasing population mortality rates. We showed that the population mortality reversal following pheromone removal was almost surely occurring in individuals, albeit more slowly than suggested by population measures. Furthermore, heterogeneity among individuals due to the inherent stochasticity of behavioural interactions skewed population mortality rates in middle-age away from the individual-level trajectories of which they are comprised. This article exemplifies how computational models function as important predictive tools for designing wet-laboratory experiments to use population mortality rates to understand how genetic and environmental manipulations affect ageing in the individual.

Keywords: ageing; heterogeneity; mortality; reproduction; stochastic models.

Figure 1.

Mortality dynamics can be explained by individual reversibility and/or population heterogeneity. (a) Increases in mortality rate from continuous pheromone exposure are proportional to the magnitude of pheromone effect, Z_n. Exposure is throughout life, with long-lasting pheromone effects (D = 35 days, equation (2.4)). (b) Late-life mortality deceleration is driven by heterogeneity in ME. Removal of heterogeneity (light blue line) restores Gompertzian dynamics reflective of ageing at the individual level. Exposure is permanent with D = 35 days and Z_n = 10^0.7. (c) Removal of pheromone exposure at day 7 leads to reversion of mortality rates to never-exposed control rates in both heterogeneous and homogeneous populations when the duration of pheromone effects is long (D = 35 days), using Z_n = 10². (d) Removal of pheromone exposure at day 7 when pheromone effects are permanent (D = 70 days) leads to reversal of mortality rates in heterogeneous, but not homogeneous, populations, using Z_n = 10². Shading in panels represent when pheromones were present.

Figure 2.

Age of pheromone removal differentiates ageing patterns. (a) Removal of pheromone exposure early in life (day 7) differentiates quickly reversible individual mortality rates from slow or non-reversible individual mortality rates. (b) When pheromones are removed in middle-age (day 30), populations with heterogeneous mortality rates can be differentiated from slowly reversible homogeneous populations. In all panels, Z_n = 10². The ‘slow’ homogeneous group is defined as D = 35, whereas the ‘quick’ homogeneous group is defined as D = 5. For the heterogeneous group effects are permanent. Shading in panels represent when pheromones were present.

Figure 3.

In vivo experiments demonstrate varying reversibility depending on age of pheromone removal. (a) When pheromones are removed at day 7, mortality rates are increased early in life, but eventually revert to those of never-exposed controls (p = 0.0002 by log-rank comparing day 7 and non-fem groups). When pheromones are removed at day 28 (b) or day 35 (c), mortality rates never completely revert to those of never-exposed controls ( by log-rank comparing day 28 or day 35 and non-fem groups). However, both the day 28 and day 35 groups are significantly longer lived than the lifelong feminized exposure group (p < 0.0001 and p = 0.012, respectively, by log-rank). N = [94 : 100] for all groups. Log-rank statistical tests compare entirety of lifespans. Shading in panels represent when pheromones were present for switch group (blue lines).

Figure 4.

In silico analysis of in vivo data reveals individual mortality rates reverse slowly. (a) McFadden's pseudo R² statistic, also known as ρ², [14] colourmap as a function of the duration of pheromone effects D (x-axis, in days) and magnitude of pheromone effects Z_n (y-axis). Outcomes represent the difference between experimental fly mortality rates data with permanent pheromone exposure and simulation outcomes for combinations of D and Z_n. Further simulations (see the electronic supplementary material, tables S1 and S2) examining best fit to the in vivo effects of pheromone removal determined that the optimal D and Z_n correspond to 35 days and 10^0.7, respectively (shown in panels b–f). In vivo mortality data closely match simulated data with a pheromone effect duration of 35 days for no exposure (b), constant pheromone exposure, (c), 7 day removal (d), 28 day removal (e) and 35 day removal (f). See the electronic supplementary material, tables S1 and S2 for fitting statistics. Shading in panels represent when pheromones were present.

Figure 5.

Heterogeneity drives late reversibility but not early reversibility. (a) When heterogeneity is removed from the simulation, the in vivo effects of early (day 7) pheromone removal are still replicated. However, when pheromone removal occurs on day 28 (b) or day 35 (c), removal of heterogeneity leads to deviation from the observed in vivo mortality rates (see the electronic supplementary material, table S3). Removal of heterogeneity does not prevent replication of never-exposed (d) or constantly exposed (e) mortality rates. Shading in panels represent when pheromones were present.