Damage segregation at fissioning may increase growth rates: A superprocess model☆
Introduction
One of the great challenges in biology is to understand the forces shaping age-related functional decline, termed senescence. Much current thinking on senescence (cf. Steinsaltz and Goldwasser, 2006) interprets the aging process as an accumulation of organismal damage. The available damage repair mechanisms fall short, it is often argued, because of limitations imposed by natural selection, which may favor early reproduction, even at the cost of later decrepitude. One line of research aims to clarify these trade-offs by examining the non-aging exceptions that test the senescence rule. Of late, it has even been argued that negligible (cf. Finch, 1990) or even negative (cf. Vaupel et al., 2004) senescence may not be as theoretically implausible as some had supposed, and that it might not even be terribly rare (cf. Guerin, 2004).
Fissioning unicellular organisms have been generally viewed as a large class of exceptions to the senescence rule. Indeed, their immortality has been considered almost tautological by the principle enunciated by Medawar (1957), that individual birth is a fundamental prerequisite for aging. This principle has been sharpened by Partridge and Barton (1993), who remark
Recent experiments (such as Aguilaniu et al., 2003, Lai et al., 2002, Ackermann et al., 2003, Stewart et al., 2005) have focused attention on the elusive quality of the “distinction” between parent and offspring. If aging is the accumulation of unrepaired damage, then “age” may go up or down. The metazoan reproduction that results in one or more young (pristine) offspring, entirely distinct from the old (damaged) parent, is an extreme form of rejuvenation. This may be seen as one end of a continuum of damage segregation mechanisms that include the biased retention of carbonylated proteins in the mother cell of budding yeast Aguilaniu et al. (2003) and perhaps the use of aging poles inherited by one of the pair of Escherichia coli daughter cells as, in the words of Stephens (2005), cellular “garbage dumps”. Even where there is no conspicuous morphological distinction between a mother and offspring, the individuals present at the end of a bout of reproduction may not be identical in age, when age is measured in accumulated damage. Whereas traditional theory has focused on the extreme case of an aging parent producing pristine offspring, it now becomes necessary to grapple with the natural-selection implications of strategies along the continuum of damage-sharing between the products of reproduction.The critical requirement for the evolution of ageing is that there be a distinction between a parent individual and the smaller offspring for which it provides. If the organism breeds by dividing equally into identical offspring, then the distinction between parent and offspring disappears, the intensity of selection on survival and reproduction will remain constant and individual ageing is not expected to evolve.
Our approach is a mathematical model of damage accumulation during a cell's lifetime and damage segregation at reproduction that quantifies (in an idealized context) the costs and benefits of unequal damage allocation to the daughter cells in a fissioning organism. The benefits arise from what Bell (1988) has termed “exogenous repair”: elimination of damage through lower reproductive success of individuals with higher damage levels.
For a conceptually simple class of models of population growth that flexibly incorporate quite general structures of damage accumulation, repair and segregation, we analytically derive the conditions under which increasing inequality in damage inheritance will boost the long-term population growth rate. In particular, for organisms whose lifetime damage accumulation rate is deterministic and positive, some non-zero inequality will always be preferred. While most immediately relevant for unicellular organisms, this principle and our model may have implications more generally for theories of intergenerational effects, such as transfers of resources and status.
One consequence of exogenous repair may seem surprising: if inherited damage significantly determines the population growth rate, and if damage is split unevenly among the offspring, there may be a positive benefit to accelerating the turnover of generations. In simple branching population growth models, the stable population growth rate is determined solely by the net birth rate. In the model with damage, increasing birth and death rates equally may actually boost the population growth rate. This may be seen as the fission analogue of Hamilton's (1966) principle linking the likelihood of survival to a given age with future mortality-rate increases and vitality decline, and placing a selective premium on early reproduction. Of course, if the variance in damage accumulation is above the optimum, then this principle implies that reducing the inequality in inheritance, or decreasing birth and death rates equally, would be favored by natural selection.
Section snippets
Background
Popular reliability models of aging (such as Koltover, 1981, Gavrilov and Gavrilova, 2001, Doubal, 1982 and additional references in Section 3 of Steinsaltz and Goldwasser, 2006) tend to ignore repair, while the class of growth-reproduction-repair models (such as Kirkwood, 1977, Abrams and Ludwig, 1995, Cichón, 1997, Mangel, 2001, Chu and Lee, 2006 and further references in Section 2 of Steinsaltz and Goldwasser, 2006) tend to ignore the fundamental non-energetic constraints on repair. A living
Description of the model
Our model for the growth of a population of fissioning organisms such as E. coli is an infinite population measure-valued diffusion limit of a sequence of finite population branching models. Mathematical derivations of important properties of this model are left for section 7 and Appendix A. In the present section, we offer a more intuitive description of the model.
Major results
Write for the asymptotic re-scaled population size at time t. That is, is the total mass of the measure . We show that there is an asymptotic population growth rate in the sense that the asymptotic behavior of the expectation of satisfiesThus, grows to first order like .
When the asymptotic growth rate is positive, this result may be significantly strengthened. Theorem 7.4 tells us then (under mild technical conditions) that:
To first order, the
An example
Suppose that damage accumulates deterministically during the life of an organism, at a rate proportional to the current level of damage. In the absence of damage segregation, then, the damage at time t grows like (this corresponds to and ). We suppose that the combined parameter describing the time rate of damage segregation for an organism that fissions at damage level x is , where is a tunable non-negative parameter (hence, ). The generator
Conclusions
We have introduced a mathematical model of population growth, for a population of haploid organisms that accumulate damage. This is a limit of branching processes, in which individuals accumulate damage, rising and falling at random, with an upward tendency. Under fairly general assumptions we have shown
The population growth rate converges to a fixed rate, determined by the solutions to an ordinary differential equation (ODE);
If the asymptotic growth rate is positive, the relative proportions
Mathematical methods
Define to be the semigroup generated by with boundary conditionfor some fixed . (This semigroup will not, in general, be sub-Markovian because can take positive values). As we observe in Section 7.2, when is the deterministic measure , and so the behavior of is governed by that of , at least at the level of expectations. The choice is most relevant for the description of fissioning
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SNE supported in part by Grant DMS-04-05778 from the National Science Foundation. DS supported in part by Grant K12-AG00981 from the National Institute on Aging and by a Discovery Grant from the National Science and Engineering Research Council.