Stochastic studies of aging and mortality in multi-cellular organisms. I. The asymptotic theory

Abstract

A quantitative phenomenological theory of aging is described, free of assumptions regarding the molecular character of initial steps in senescence, yet consistent with known kinetic and thermodynamic results. As such, it becomes a yardstick against which microscopic theories must eventually be judged. The derivation is couched in terms of the operation of an abstract biological clock whose role is to count the occurrence of k otherwise unspecified rate-limiting events out of a larger set of n. We establish the counting rules by which such a clock may operate, and discover that observed kinetic results have their fundamental explanation in general probabilistic considerations. If the rate-limiting events occur in random order, and n is permitted to grow large, then for k = 1 only three nonperiodic asymptotically stable, limit distributions exist to describe the survival of a cohort as a function of time. One solution yields the power-law expression for the mortality rate; the second gives Gompertz law; and the third is of no value in the present context. These results are obtained without reference to the mathematical character of possible rate-limiting processes. In fact, for arbitrarily chosen functions, no asymptotically stable limit distributions generally exist. Criteria are provided which define the domains of attraction for the two limit distributions of interest. This makes it possible to determine whether a suggested rate-limiting process would be consistent with the power- law, Gompertz law or neither, and obviates the need for detailed and time-consuming mathematical analysis. It becomes clear that the mathematical character of the rate- limiting process may differ sharply from the power law or Gompertz law yet, by virtue of the counting rule established for the clock's operation, still yield the macroscopic kinetic results observed in practice. The consequence of this finding is that macroscopic kinetic results do not reflect, in any simple way, the underlying molecular mechanisms responsible for senescence. In fact, no rate-limiting process may be uniquely chosen on the basis of kinetic results alone.