October 24, 2022
There has been a recent surge of interest in what causes aging. This has been matched by unprecedented research investment in the field from tech companies. But, despite considerable effort from a broad range of researchers, we do not have a rigorous mathematical theory of programmed aging. To address this, we recently derived a mortality equation that governs the transition matrix of an evolving population with a given maximum age. Here, we characterize the spectrum of eigenvalues of the solution to this equation. The eigenvalues fall into two classes. The complex and negative real eigenvalues, which we call the flower, are always contained in the unit circle in the complex plane. They play a negligible role in controlling the dynamics of an aging population. The positive real eigenvalues, which we call the stem, are the only eigenvalues which can exceed the unit circle. They control the most important properties of the dynamics. In particular, the spectral radius increases with the maximum allowed age. This suggests that programmed aging confers no advantage in a constant environment. However, the spectral gap, which governs the rate of convergence to equilibrium, decreases with the maximum allowed age. This opens the door to an evolutionary advantage in a changing environment.
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Aging is thought to be a consequence of intrinsic breakdowns in how genetic information is processed. But mounting experimental evidence suggests that aging can be slowed. To help resolve this mystery, I derive a mortality equation which characterizes the dynamics of an evolving population with a given maximum age. Remarkably, while the spectrum of eigenvalues that govern the evolution depends on the fitness, how they change with the maximum age is independent of fitness. Thi...
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The chronological age used in demography describes the linear evolution of the life of a living being. The chronological age cannot give precise information about the exact developmental stage or aging processes an organism has reached. On the contrary, the biological age (or epigenetic age) represents the true evolution of the tissues and organs of the living being. Biological age is not always linear and sometimes proceeds by discontinuous jumps. These jumps can be positive...
September 3, 1997
We present a model for biological aging that considers the number of individuals whose (inherited) genetic charge determines the maximum age for death: each individual may die before that age due to some external factor, but never after that limit. The genetic charge of the offspring is inherited from the parent with some mutations, described by a transition matrix. The model can describe different strategies of reproduction and it is exactly soluble. We applied our method to...
May 24, 2013
What is aging? Mechanistic answers to this question remain elusive despite decades of research. Here, we propose a mathematical model of cellular aging based on a model gene interaction network. Our network model is made of only non-aging components - the biological functions of gene interactions decrease with a constant mortality rate. Death of a cell occurs in the model when an essential gene loses all of its interactions to other genes, equivalent to the deletion of an ess...
April 26, 2005
In these lecture notes I describe some of the main theoretical ideas emerged to explain the aging dynamics. This is meant to be a very short introduction to aging dynamics and no previous knowledge is assumed. I will go through simple examples that allow one to grasp the main results and predictions.
November 12, 2010
Aging is a fundamental aspect of living systems that undergo a progressive deterioration of physiological function with age and an increase of vulnerability to disease and death. Living systems, known as complex systems, require complexity in interactions among molecules, cells, organs, and individuals or regulatory mechanisms to perform a variety of activities for survival. On this basis, aging can be understood in terms of a progressive loss of complexity with age; this sug...
The principle of linearized stability and instability is established for a classical model describing the spatial movement of an age-structured population with nonlinear vital rates. It is shown that the real parts of the eigenvalues of the corresponding linearization at an equilibrium determine the latter's stability or instability. The key ingredient of the proof is the eventual compactness of the semigroup associated with the linearized problem, which is derived by a pertu...
We present an individual based model of evolutionary ecology. The reproduction rate of individuals characterized by their genome depends on the composition of the population in genotype space. Ecological features such as the taxonomy and the macro-evolutionary mode of the dynamics are emergent properties. The macro-dynamics exhibit intermittent two mode switching with a gradually decreasing extinction rate. The generated ecologies become gradually better adapted as well a...
April 16, 2020
We study a mathematical model describing the growth process of a population structured by age and a phenotypical trait, subject to aging, competition between individuals and rare mutations. Our goals are to describe the asymptotic behaviour of the solution to a renewal type equation, and then to derive properties that illustrate the adaptive dynamics of such a population. We begin with a simplified model by discarding the effect of mutations, which allows us to introduce the ...
Well protected human and laboratory animal populations with abundant resources are evolutionary unprecedented, and their survival far beyond reproductive age may be a byproduct rather than tool of evolution. Physical approach, which takes advantage of their extensively quantified mortality, establishes that its dominant fraction yields the exact law, and suggests its unusual mechanism. The law is universal for all animals, from yeast to humans, despite their drastically diffe...