Programed Death is Favored by Natural Selection in Spatial Systems

Justin Werfel, Donald E. Ingber, and Yaneer Bar-Yam

Phys. Rev. Lett. 114, 238103 – Published 12 June 2015

Abstract

Standard evolutionary theories of aging and mortality, implicitly based on mean-field assumptions, hold that programed mortality is untenable, as it opposes direct individual benefit. We show that in spatial models with local reproduction, programed deaths instead robustly result in long-term benefit to a lineage, by reducing local environmental resource depletion via spatiotemporal patterns causing feedback over many generations. Results are robust to model variations, implying that direct selection for shorter life span may be quite widespread in nature.

Received 18 April 2013

DOI:https://doi.org/10.1103/PhysRevLett.114.238103

Authors & Affiliations

Justin Werfel^1,2,3,*, Donald E. Ingber^2,3,4,†, and Yaneer Bar-Yam^1,‡

¹New England Complex Systems Institute, 210 Broadway, Suite 101, Cambridge, Massachusetts 02139, USA
²Wyss Institute for Biologically Inspired Engineering, Harvard University, Cambridge, Massachusetts 02138, USA
³Harvard Medical School and Children’s Hospital, Boston, Massachusetts 02115, USA
⁴School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

^*justin.werfel@wyss.harvard.edu
^†don.ingber@wyss.harvard.edu
^‡yaneer@necsi.edu

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Vol. 114, Iss. 23 — 12 June 2015

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Figure 1
Longer life span can give short-term advantage and long-term disadvantage in spatial models, in contrast to predictions based on mean-field approximation. (a)–(c) Plots show, for invaders with life span $L$ introduced into a steady-state population with equilibrium life span $L = L_{0}$ , reproductive success (average number of consumers of the introduced type present) as a function of time. (Dashed) Mean-field calculations give the incorrect prediction that longer life span is always favored. (Solid) Numerical results from the spatial model show that longer life span can be favored for periods of hundreds or thousands of time steps but outcompeted in the long term. The time scale of the simulated spatial dynamics is also much longer than the mean-field result [29]. Parameter values are (a) $g = 0.13, v = 0.2, L_{0} = 2.7$ , (b) $g = 0.17, v = 0.1, L_{0} = 7.6$ , (c) $g = 0.2, v = 0.005, L_{0} = 170$ , with resource growth $g$ , consumption $v$ , and reproduction cost $c = 0$ . Error bars show the standard error of the mean among ten independent runs, each recording the mean for a set of 40 000 invasions. (d),(e) Longer-lived consumers deplete their local environment compared to shorter-lived ones. (d) Effective contact rate $ρ^{'} = \sum_{j \in n n in state R} [1 / N_{C} (j)]$ , averaged over all invaders, as a function of time since introduction. A consumer’s $ρ^{'}$ is the number of neighboring resource sites, corrected for the number of their neighbors that might consume them [23]. The local environment changes from a value characteristic of the invaded population to a value characteristic of the invader within a few dozen time steps. (Data for high $q$ are more variable since the invaders are quickly outcompeted and so fewer samples are available.) (e) Expected lifetime reproduction $ρ^{'} p / (v + q)$ . (Solid) Numerical results, based on the simulated $ρ^{'}$ [(d), averaged over time steps 51–100] where the invader has transformed its local environment, show a disadvantage for invaders with life span longer or shorter than the equilibrium value of 0.215. Error bars show standard error of the mean. (Dashed) Mean-field calculations for invasion of an infinite population, with no local effect on environment, incorrectly predict an advantage for invaders with longer life span. Parameter values are $g = v = 0.1, c = 0, p = 0.9991$ .
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Figure 2
Ascendance studies favor intrinsic mortality in numerical simulations with a spatial model. (a) Snapshots showing different spatial distributions of resources (yellow) and immortal (left, magenta) or mortal (right, cyan) consumers ( $50 \times 50$ subsets of $250 \times 250$ lattices). (b) History of evolving consumer intrinsic mortality $q$ in one example trial ( $g = v = 0.1, c = 0, μ_{p} = μ_{q} = 0.01$ ). Mean-field analysis (red, dashed; arrow) predicts mean $q$ quickly goes to 0. Numerical simulations (blue, solid; population mean-maximum-minimum) show long-term stability of finite $q$ and elimination of low- $q$ strains from the population. (c),(d) Steady-state average values of (c) consumer reproduction probability $p$ and (d) intrinsic life span $L = 1 / q$ , for different values of parameters $g$ and $v$ and for (c),(d) mortal (solid) and (c) immortal (dashed) populations. (e) Steady-state evolved $q$ in mortal populations ( $g = v = 0.05, c = 0$ ) for increasing neighborhood size $γ$ . For high enough $γ$ , consumers with $q = 0$ are not eliminated from the population. (f) Steady-state evolved $q$ for the mortal populations of (d) approximately falls on a single curve [line is a power-law fit: $q = 0.245 (g / v)^{- 1.07}$ ] when plotted as a function of “consumption ratio” $g / v$ , for all $g$ and $v$ tested. All error bars show the standard error of the mean from ten independent trials.
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Figure 3
A successful invasion of immortal consumers by mortal ones. (a) The fraction of invaders in the population (solid line) increases almost monotonically with time. The region dominated by mortals grows steadily: the dashed line shows the area of a circle (under periodic boundary conditions) whose radius increases at a constant rate (correlation $r = 0.997$ ). Resource growth $g = 0.05$ , consumer consumption $v = 0.2$ , consumer reproduction cost $c = 0$ . (b) Snapshots at 50, 1350, and 2550 time steps [colors as in Fig. 2].
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