Optimal Lineage Principle for Age-Structured Populations

Abstract

We present a formulation of branching and aging processes that allows age distributions along lineages to be studied within populations, and provides a new interpretation of classical results in the theory of aging. We establish a variational principle for the stable age distribution along lineages. Using this optimal lineage principle, we show that the response of a population's growth rate to age-specific changes in mortality and fecundity-a key quantity that was first calculated by Hamilton-is given directly by the age distribution along lineages. We apply our method also to the Bellman-Harris process, in which both mother and progeny are rejuvenated at each reproduction event, and show that this process can be mapped to the classic aging process such that age statistics in the population and along lineages are identical. Our approach provides both a theoretical framework for understanding the statistics of aging in a population, and a new method of analytical calculations for populations with age structure. We discuss generalizations for populations with multiple phenotypes, and more complex aging processes. We also provide a first experimental test of our theory applied to bacterial populations growing in a microfluidics device. K E Y W O R D S :Age structure, fecundity, models/simulations, population structure, selection, senescence.The age structure of populations has been studied intensively both theoretically and experimentally over the past century (Charlesworth 1994). Recent work on aging in bacteria and in single-celled eukaryotes has brought the powerful techniques of microbiology to bear on aging studies (Ackermann et al. 2003; Stewart et al. 2005; Wakamoto et al. 2005;Ackermann et al. 2007;Lindner et al. 2008;Ping et al. 2008;Erjavec et al. 2008;Shcheprova et al. 2008; Rokney et al. 2009; Wang et al. 2010). In particular, study of aging at the single-cell level, using microfluidics and automated image analysis, is allowing the interplay between aging and population dynamics to be examined at a new level of detail (Wakamoto et al. 2005; Wang et al. 2010). The entire lineage structure of a population that exhibits aging has been measured in several recent studies (Lindner et al. 2008; Stewart et al. 2005).These experiments provide a wealth of information, but their interpretation is not straightforward despite the multitude of known theoretical results regarding the dynamics of agestructured populations. Most theoretical studies of age structure tend to focus on population age distributions, and are not geared to describe the distribution of ages along individual lineages, and until the very recent work on bacterial aging (Lindner et al. 2008; Stewart et al. 2005), such data have not generally been available. In fact, as we show in this work, proper attention to the statistical structure of lineages not only provides the missing framework needed for data interpretation, but also fills gaps in understanding of the age-structured population theory itself. Our method of lineage analysis simultaneous...

Noise-driven growth rate gain in clonal cellular populations

“…Most studies on cellular growth have referred to bulk population growth rates in measurements, but recent research on the heterogeneous and stochastic nature of growth at the single-cell level poses new, fundamental questions regarding (i) how precisely the insights gained through population measurements apply to single-cell properties (3), and (ii) how single cells as a whole bring about the observed population properties (12,13). Our study tackled the latter issue both experimentally and theoretically, showing that a simple age-structured population model explains the intricate relationships between the growth rates of the cells and the population.…”

“…Noting that g * ðτÞ is the typical lineage generation time distribution within a population, we can understand from the above relations that growth rate differences are linked with the different statistics along cell lineages in isolation and within a population. The advantage of the lineage viewpoint in analyzing population dynamics has been explored theoretically (12,(25)(26)(27)(28); our study provides an experimental foundation behind this powerful concept.…”

“…To gain further insights into the observed growth rate gain, we compared the experimental results with the prediction of an age-structured population model (Fig. 3D) (12,13,24) in which cells divide according to an agedependent division probability. An important prediction of this model is that the generation time distribution g(τ) and population growth rate Λ p must satisfy the so-called Euler-Lotka equation:…”

“…1). Intrapopulation competition complicates single-cell analysis because any growth-correlated quantities measured over the population deviate from intrinsic single-cell properties (10)(11)(12). In the case of the toy model described in Fig.…”

Analysis of Cell Size Homeostasis at the Single-Cell and Population Level

“…The weights (1b) follow because there are D j measurements in the jth lineage and N L lineages. Such a lineage can be thought of choosing a random cell of the final population and traversing the tree from leave to root as described in Nozoe et al [29], Wakamoto et al [30], and Thomas [31].…”