Flowers of immortality Thomas Fink and Yang-Hui He London Institute for Mathematical Sciences Royal Institution 21 Albermarle St

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Flowers of immortality

Thomas Fink and Yang-Hui He

London Institute for Mathematical Sciences, Royal Institution, 21 Albermarle St,

London W1S 4BS, UK

Abstract. 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 lie outside 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.

arXiv:2210.13561v1 [q-bio.PE] 24 Oct 2022

Flowers of immortality 2

1. Introduction

1.1. The mortality equation

Recently, we derived a simple mortality matrix equation that governs the transition

matrix Qof an evolving population with maximum age a[1]. It is

Qa(I+MF −Q) = MF,(1)

where all of the matrices are 2n×2nand Fis a diagonal matrix with the genotype

ﬁtnesses along the diagonal. The mutation matrix Msatisﬁes

Mn=Mn/n, (2)

where Mis deﬁned recursively in block form:

Mn+1 = MnIn

InMn!,M1= 0 1

1 0 !,(3)

with Inthe 2n×2nidentity matrix. The only diﬀerence between Mand Mis that M

is a stochastic matrix, in that its rows and columns sum to 1.

Eq. (1) is actually a more compact version of

Qa=MF(I+Q+. . . +Qa−1),(4)

which is the actual governing equation. Notice that while eq. (1) has degree a+ 1, it

can be reduced to degree aby dividing through by Q−Ito give eq. (4). The solution

Q=Ito eq. (1) is spurious, and we always disregard it. For the derivation of eq. (4)

and (1), see [1].

The matrix Q=MF governs the evolution of a population with maximum age

a= 1. It has 2neigenvalues, some of which may be degenerate. They are all real and lie

in the range [−1,1].

By the Cayley Hamilton theorem, the eigenvalues of Qhave the same functional

relation to the eigenvalues of MF as the matrix Qdoes to the matrix MF. For each

eigenvalue λof MF, we obtain a family of aeigenvalues belonging to Q. This family is

generated by the analogue of eq. (1), but for numbers rather than matrices:

µa(1 + λ−µ) = λ. (5)

Just as eq. (1) is more the compact version of eq. (4), albeit with a spurious root at

Q=I, eq. (5) is a more compact version of

µa=λ(1 + µ+µ2+. . . +µa−1),(6)

with a similarly spurious root at µ= 1, which we always disregard. Eq. (6) has a

solutions, and thus there are a2neigenvalues of the matrix Q. The goal of this paper is to

characterize them, and thereby better understand the dynamics of an aging population.

Notice that while the eigenvalues λare real, the eigenvalues µcan be complex.

Whereas a real eigenvalue corresponds to exponential growth or decay of the component

Flowers of immortality 3

of the population that projects along the associated eigenvector, a complex eigenvalue

corresponds to oscillatory behavior, with the overall growth or decay set by its

magnitude.

1.2. Summary of results

In this paper, we do four things, which correspond to the following four sections.

In the section 2, we study the complex and negative real eigenvalues of the solution

Qto the mortality equation (1). Our approach is make no assumptions about the ﬁtness

F, and therefore about the 2neigenvalues λ∈[−1,1] of MF. Rather it is to characterize,

for any given λ, the family of aeigenvalues satisfying eq. (5). The complex and negative

real eigenvalues are always contained in the unit circle, which we call the ﬂower (Fig.

1). These play a negligible role in controlling the dynamics of an aging population.

In the section 3, we study the real eigenvalues of Q, which we call the stem. Of the

aeigenvalues associated with a given λ, only one is positive and real, which we call ρ,

and it has the largest magnitude (Fig. 1). Only these stem eigenvalues can lie outside

the unit circle. They control most of the important properties of the dynamics of an

aging population.

We study properties of the stem in section 4. We prove that the stem eigenvalues

ρincrease with λand are convex with λ, and increase with the maximum age a. The

latter is important because it means that, in a ﬁxed environment, programmed aging

confers no evolutionary beneﬁt. We show that the spectral gap of the eigenvalues of Q

increases as adecreases. This is important because the rate of convergence to equilibrium

is controlled by the spectral gap [2].

In section 5, we test our predictions by calculating the actual eigenvalues for two

diﬀerent ﬁtness functions: uniform ﬁtness and Hamming ﬁtness. Both are plotted in Fig.

3 and perfectly agree with our predictions. We conclude with a discussion, where we

describe the implications of our results on programmed aging.

2. The ﬂower: complex and negative real eigenvalues

In this section we consider the complex and negative real eigenvalues of the solution of

the mortality equation. We call this the ﬂower, because of the petals that appear as the

maximum age aincreases (Fig. 1). We show that all these eigenvalues are tame, in the

sense that they are contained inside the unit circle on the complex plane.

Our protagonist is the polynomial from eqs. (5) and (6), recalling that the more

compact P= (µ−1)Qhas an (a+ 1)-th spurious root at µ= 1:

P(µ) := −λ+ (1 + λ)µa−µa+1,(7)

Q(µ) := λ(1 + µ+µ2+. . . +µa−1)−µa.(8)

This canonical form makes it easier to pick oﬀ the coeﬃcients in what follows. We ﬁrst

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FlowersofimmortalityThomasFinkandYang-HuiHeLondonInstituteforMathematicalSciences,RoyalInstitution,21AlbermarleSt,LondonW1S4BS,UKAbstract.Therehasbeenarecentsurgeofinterestinwhatcausesaging.Thishasbeenmatchedbyunprecedentedresearchinvestmentintheeldfromtechcompanies.But,despiteconsiderableeortfrom...

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Flowers of immortality Thomas Fink and Yang-Hui He London Institute for Mathematical Sciences Royal Institution 21 Albermarle St

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