The evolution of senescence in multi-component systems

Actuarial senescence is characterized by an increase in mortality rate with increasing chronological age. The reliability theory of senescence proposes that organisms' vital functions can be modelled as a suite of damageable, irreplaceable elements (typically genes or their products) that protect their bearer from condition-dependent death so long as at least one of the elements remains intact. Current incarnations of the reliability theory of senescence are continuous-time models with no explicit evolutionary component. Here, we use elementary probability theory and evolutionary dynamics analysis to derive a discrete-time version of the reliability theory of senescence. We include three variations on this theme: the 'Series' model in which damage to any of n elements results in death, the 'Parallel' model, in which damage accumulates in random order and damage to all n elements results in death, and the 'Cascade' (multi-stage) model, which is like the Parallel model, except the irreparable damage necessarily follows a strict sequence. For simplicity, we refer to the state of having multiple elements as 'redundancy', but this does not imply that the elements are necessarily identical. We show that redundancy leads to actuarial senescence in the Parallel and Cascade models but not in the Series model. We further demonstrate that in the Parallel and Cascade models, lifetime reproductive output (a potential proxy for fitness in populations with discrete generations) is a positive but decelerating function of redundancy. The positive nature of the fitness function leads to the prediction that redundancy and senescence should evolve from non-redundant, non-senescing ancestral populations; however, the deceleration of the fitness function leads to the prediction that this evolution towards increased redundancy will eventually be limited by mutation-selection balance. Using evolutionary dynamics analysis involving the discrete-generation quasispecies equation, we confirm these two predictions. Finally, we show that a population's equilibrium redundancy is sensitive to the environmental conditions that prevailed during its evolution, such as the rate of extrinsic mortality.