A pace and shape perspective on fertility

Ageing is ubiquitous to all organisms, but ageing does not always mean senescence. Counter to most evolutionary theories of ageing, the patterns of mortality and reproduction may remain unchanged or improve with age, as well as deteriorate. Describing this diversity presents a challenge to eco?evolutionary demography. The pace–shape framework of mortality tackled this challenge to qualify and quantify orthogonal components of ageing patterns in mortality. Here, we extend this framework to fertility.Analogous to the logic of the mortality framework, we define a perspective, a framework and novel methods for the pace and shape of fertility. These distinguish between orthogonal components of time?scale (pace) and distribution (shape) of reproduction over adult life span.Our pace and shape framework mirrors that of mortality, through a shift of perspective from the mother giving birth, to the offspring being born. Our new measures overcome many problems associated with measuring natural fertility trajectories, have both a clear biological and mathematical interpretation, can be intuitively visualized and satisfy and extend important conditions of the pace–shape paradigm.A comprehensive framework of fertility pace–shape facilitates ecological and evolutionary research addressing interactions and trade?offs between components of birth and death patterns, across the whole tree of life. The burgeoning emergence of large comparative demographic data sources across wide environmental, geographical, temporal and phylogenetic ranges, combined with pace–shape measures, opens the door to comparative analyses of ageing which were never possible before.</p

Fixation probability as a function of post-reproductive lifespan.

<p>A density-independent, stationary, closed-to-migration population of size <i>N</i> was initiated with a single mutant with advantage <i>s</i> in the first age class. At each simulation step, all individuals in age classes smaller than or equal to 2 produce exactly one newborn each. Individuals in the maximum age class (i.e. 2 + post-reproductive ages) are removed from the population. Resident and mutant individuals survive to the next age class with probability 0.618034, but while resident newborns enter the first age class with the same probability, mutant newborns do so with probability 0.618034 + <i>f</i>(s), where <i>f</i>(s)>0 is such that the mutant growth rate is equal to <i>s</i> plus the resident growth rate. The simulation run ended with population extinction or fixation of one of the two types. The population was considered extinct when there was no individual with reproductive value. Fixation for one type was considered achieved when the total reproductive value in the population was exclusively contributed to by that type’s subpopulation. Fixation probability of the mutant was calculated as the number of times it achieved fixation in 100000 simulation runs without considering those runs in which the whole population went extinct before fixation of either type. We compared simulation results (solid line) with analytic results (dashed line) derived from the approximation in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0133820#pone.0133820.e011" target="_blank">Eq (4)</a> in the main text. We explored the role of increasing the maximum attainable age, the total population size, and the magnitude of the advantageous effect of the mutation.</p

A pace and shape perspective on fertility

Ageing is ubiquitous to all organisms, but ageing does not always mean senescence. Counter to most evolutionary theories of ageing, the patterns of mortality and reproduction may remain unchanged or improve with age, as well as deteriorate. Describing this diversity presents a challenge to eco?evolutionary demography. The pace–shape framework of mortality tackled this challenge to qualify and quantify orthogonal components of ageing patterns in mortality. Here, we extend this framework to fertility.Analogous to the logic of the mortality framework, we define a perspective, a framework and novel methods for the pace and shape of fertility. These distinguish between orthogonal components of time?scale (pace) and distribution (shape) of reproduction over adult life span.Our pace and shape framework mirrors that of mortality, through a shift of perspective from the mother giving birth, to the offspring being born. Our new measures overcome many problems associated with measuring natural fertility trajectories, have both a clear biological and mathematical interpretation, can be intuitively visualized and satisfy and extend important conditions of the pace–shape paradigm.A comprehensive framework of fertility pace–shape facilitates ecological and evolutionary research addressing interactions and trade?offs between components of birth and death patterns, across the whole tree of life. The burgeoning emergence of large comparative demographic data sources across wide environmental, geographical, temporal and phylogenetic ranges, combined with pace–shape measures, opens the door to comparative analyses of ageing which were never possible before.</p

First row, age-specific mortality curves for C. elegans, humans, <i>D. melanogaster</i> and yeasts.

<p>The qualitative pattern goes from an exponential-exponential pattern to an exponential-U-shape-exponential pattern. Experimental data adapted from Vaupel et al. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002825#pcbi.1002825-Vaupel1" target="_blank">[7]</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002825#pcbi.1002825-University1" target="_blank">[6]</a> for humans.; Second row: Mortality patterns for an individual mortality function (HRM) in different environments (, 500 individuals per simulation, 400 generations and 300 simulations). The same transitions as the one depicted in the first row; Third row: Mortality patterns for a mortality function (HTM) resulting from the evolution under different values for (, 500 individuals per simulation, 400 generations and 300 simulations per figure). The same transitions as in the first two rows are observed, from an exponential pattern to an exponential-U-shape-exponential pattern.</p

Survival curves after a heat-shock with heterogeneity in aging rates (HRM) and heterogeneity in aging timing (HTM), on the right column.

<p>The two curves reproduce the main features of survival curves corresponding to heat-shock experiments in C. elegans (filled diamonds, adapted from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002825#pcbi.1002825-Wu1" target="_blank">[24]</a>). Both the experimental data and the simulated curves exhibit a two stage decrease: a first quick, strong fall followed by a slowing down. (, 500 individuals per simulation, 500 simulations per curve).</p

Population heterogeneity depends on the interactions between the species biology and its environment.

<p>These interactions are captured by the parameter . The resulting distributions of heterogeneity for an exponential frailty model are presented above, corresponding to the mortality curves presented in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002825#pcbi-1002825-g001" target="_blank">figure 1</a>, second row. represents the investment in reproduction made by the individual. Higher basal damage accumulation (high ) leads to more individuals investing in maintenance (i.e., low due to life-history trade-off). In both this figure and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002825#pcbi-1002825-g004" target="_blank">figure 4</a>, the scale of the y-axis is set to match the highest frequency.</p

Spearman’s rank correlation coefficients.

<p>Spearman’s rank correlation coefficients of the values of shape measures <i>S</i>₁–<i>S</i>₇ applied to the data of ten populations taken from [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0119163#pone.0119163.ref019" target="_blank">19</a>].</p><p>Spearman’s rank correlation coefficients.</p

Shape values for <i>μ</i>₁ and <i>μ</i>₂.

<p>Shape values assigned to <i>μ</i>₁ and <i>μ</i>₂ by the measures <i>S</i>₁–<i>S</i>₇. Note that the shape values assigned to <i>μ</i>₂ are consistently smaller than the shape values assigned to <i>μ</i>₁, which classify population two as showing weaker aging than population one.</p><p>Shape values for <i>μ</i>₁ and <i>μ</i>₂.</p

Expected log-mortality patterns in stress induction experiments for the HTM.

<p>Modifying without adaptation in population heterogeneity changes the mortality pattern. The results presented in the right column are in agreement with experimental results concerning diet changes in C. elegans <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002825#pcbi.1002825-Baeriswyl1" target="_blank">[9]</a>.</p

Species in pace-shape space.

<p>Pace values (as measured by life expectancy <i>e</i>₀) on the x-axis versus shape values (as measured by <i>S</i>₇) on the y-axis for ten populations taken from [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0119163#pone.0119163.ref019" target="_blank">19</a>] (HG stands for hunter-gatherers, ChimpW for wild chimpanzees, ChimpC for captive chimpanzees).</p