Senescence from Reproduction

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Submitted on 1 Jan 1995

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Senescence from Reproduction

John Thoms, Peter Donahue, Naeem Jan

To cite this version:

John Thoms, Peter Donahue, Naeem Jan. Senescence from Reproduction. Journal de Physique I,

EDP Sciences, 1995, 5 (8), pp.935-940. �10.1051/jp1:1995173�. �jpa-00247117�

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Classification Physics Abstracts

02.50-r 05.20-y 87.10+e

Short Communication

Senescence ftom Reproduction

John

Thoms,

Peter Donahue and Naeem Jan

Physics Department, St. Francis Xavier University, Antigonish, Box 5000, Nova Scotia, B2G 2W5, Canada

(Received

2 June 1995, accepted 7 June

1995)

Abstract. We present ^results ^from ^Monte ^Carlo simulations of _a modified

"bit-string"

model of Penna which show that ageing _or _senescence _may be the result of reproduction over an

extended period, _as expected from mutation accumulation. This mduces _an effective form of

antagonistic pleiotropy. We also obtain _a double maxima in the distribution of the _age at which death _occurs. These results may be relevant to current theories of ageing.

The

understanding

of the

mechanism(s)

of

ageing

or senescence is _one which is not without interest to

biologists

and others.

Ageing

is the term

commonly

used to describe the

dedining

functional

capacities

of the mature

organism.

This

problem, although

mentioned in the

Bible,

was

recognised

in the scientific literature _as

early

_as 1870

by

Russel

Wallace,

^and ^his ^theme

was

developed

later

by Weismann, Haldane, Hamilton,

Medawar and others

Ill.

This

theory

asserts that

ageing

is the result of late

acting

deleterious mutations which _are

weakly

selected

by

natural selection at

reproduction.

The _presence of these "bad" mutations in the individual _are

expressed

late in life and decrease the

ability

of the _organism _to _carry _out its normal

functions;

this makes the

organism

_more

susceptible

to

injury,

disease and then death. Another

theory

assumes that genes which _are beneficial to the

juvenile

_may turn out to be detrimental iii the

adult. This

theory

is referred _to iii the literature _as

autagonistic pleiotropy

aud _was considered

in various forms

by

Darwin,

Medawar, Williams,

Charlesworth and others

Ill.

We _propose to

study

the consequences of _a model of mutation accumulation. The

"bit-string"

model of Penna [2] is a suitable

starting point.

In this model _we have 64 genes,

representing

an

individual,

which _may be either "on" _or "off"

(0

_or

1).

At the start _we generate a random sequence of states,

representing optimal

fitness for the environment, ^which we refer _to _as the

optimal

_sequence. Dur initial

population

consists of 20000 individuals half of whom bave the

optimal

_sequence. This _was _necessary to ensure that there _was

a

surviving

_group alter the first 10 years. In this _version of the model _we do trot allow for survival _over trie _age of 64, but with trie parameters ^considered _very ^few ^live over trie _age of 35 anyway. Each

organism,

at

birth,

is

given

the

genetic string

^of its parent, ^but we

allow,

with

equal probability

for either

0,

1 _or

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936 JOURNAL DE PHYSIQUE I N°8

2

hereditary point

mutations at birth. This _is done

by simply switching

_a

randomly

selected gene of the parent from its present state to the

opposite

state.

Given _a

population

_we increment the age, 1, of each individual in the

population by

_one

la year)

and this is

represented by

the

expression

of the i~h _gene. The

newly expressed

_gene is

compared

to its counterpart ⁱⁿ ^the

optimal

_sequence. If

they

_agree, then this represents a

"good"

_gene

but if

they disagree,

then this is considered _a deleterious

hereditary

mutation. All the earlier

expressed

_genes _are also active and the total deleterious

hereditary

mutations is the

Hamming

distance between the active segment ^of ^the

genetic string

and the

corresponding

segment of the

optimal

_sequence. Also at each _year _we allow each member to

acquire

_a deleterious somatic mutation. A deleterious somatic mutation is _one which is detrimental to the

organism

but is not transmitted _to its

off-spring.

This mutation _occurs with

probability

0.01 in each _year and trie

sum of these mutations is retained. After each member of trie

population

bas

aged

_we _measure

the _average number of deleterious mutations _per individual. This average is determined from the total deleterious mutations which has occurred to ail the

organisms

_so far in their life

(hereditary plus somatic).

We next check the members of trie

population

for survival. There _are several hurdles: We compute _a ^death

probability,

_pd

~~

exp(r(b a))

₊ 1 ~~~

where b is trie _average number of mutations per individual and _a _is the number of mutations of trie individual concerned and _r, which is set

equal

to 10 in this

work, plays

trie role of _an inverse

temperature ^for _our ^Fermi ^function _[3]. ^This ^value ^of r

gives

a low

probability

of survival for those individuals below the _average fitness and

corrspondingly

_a

high probability

of survival for those whose fitness is greater than b. If _a random number is less than _pd then trie

organism

is removed from the

population.

Environmental limitations

placed

_on

food,

_space, and

nesting

sites _are taken into account

by

_an _age

independent

Verhulst term which

gives

each invididual

a

probability

of

il N(t)/k)

of

staying

alive. k is the carrying

capacity

of the environment which is set to

100,000

at the onset, ^and

N(t)

is the

population

at the year t. k is about 10 times

langer

than trie

steady

state

population

which is

approximately

8500. Trie

surviving

members _are those who _are able to avoid these _two hurdles.

After the

grim

_reaper we consider birth. Each member of the

population

in the age group ii to 25 is

given

_a 50 percent

probability

to

give

birth to _an

off-spring.

The

off-spring

is

given

_more

or less the _same

genetic

_sequence _as the parent. ^The maximum number of point mutations allowed in the

baby

is 2 and _as _we have

described,

this

simply

_means that _a maximum of 2 genes

are switched from their present state. No further

hereditary point

mutations _are allowed for this individual

during

its life time, ^thus ^all ^other deleterious mutations that _occur _are somatic.

We _now discuss the results from

simulating

this model. We have monitored the

population

to establish that _we have reached the

steady

state.

Figure

i shows the

steady

state age distribution after 2000 years. Similar _curves _are obtained alter 1000 _years. Note the

exponential decay

and also that very few members survive after the age of 25.

Figure

2 shows the average number of

deletenous mutations in the

steady

state

population

with _age. We

clearly

_see that the _average number of mutations _in the

juvenile (less

thon

10)

is very

small,

whereas the _average number of deleterious mutations increases

rapidly

after the _age of là- We _measure the average number of deleterious

hereditary

mutations per individual between the _ages of I and 12 and report 0.157,

whereas,

between the _ages of13 and 25 _we find 1.712.

Figure

3 shows the

frequency

at which death _occurs with age. We collected these data _over 1000 years in the

steady

state. There _are two

peaks

_one _near the age of 5 and the other _near

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AGE DISTRIBUTION

900

800 ^°

700

> 6°° ^°

O °

~z

500 ^°

Q _o

~

~ 400 °

o o

300 ^°

o o

200 _o

°

o o

100 o

~ °

o o

~ o

0 5 10 15 20 25 30 35

AGE

Fig. i. The _age distribution of the population at 2000 years.

the age of 21. As far _as _we _are _aware, this is the first report ^{of such} _a curve from theoretical considerations of _a model of

ageing.

^As

expected, (Fig. I),

_very few individuals survive after the _age of 25. In order _to understand the _nature of the first

peak

_we show _in

Figure

4 the

number of mutations in each _age _group. We have not normalised

by

the number of members

in each age group. We _see that the

interplay

between the

population

distribution with age and the

probability

of

expressing

_a deleterious mutation, which increases with age, is

responsible

for the first

peak.

The second

peak

reflects the _upper limit of selection at the onset of

infertility.

Here _we _sec clear demonstration of the mutation accumulation without selection _pressure trie individuals become unfit. This is reinforced in

Figure

5 where _we show trie survival

ratio,

N(t)/(N(t I)).

Here _we _see _a _more _or less constant survival ratio up ^to trie _age of 15 and this decreases to

approximately

o-S at age 25. This _is trie random limit where trie

probability

of _a deleterious mutation is o-S in the

steady

state

population.

We have shown that the Penna

"bit-string"

model is well suited for

simulating

the consequences of the mutation accumulation

theory

of

ageing.

^We ^have ^included a fitness function which

is

responsive

to trie overall fitness of trie

population.

This is _a natural method to include trie Red

Queen

_[4] effect _as the

population

evolves _to

higher

fitness. We _are also able to

impose environmental tolerance to _an individual's deviation from the average fitness. This is

accomplished by simply tuning

the _r parameter _in our Fermi function. We observed the total number of deleterious mutations per individual under trie _age of12 and between trie _ages of 13 and 25 and find that it differs

by

_a factor of 10. We _may

interpret

this _as

reflecting

_a type of

antagonistic pleiotropy.

^Trie

juvenile

is

simple

fitter _on average thon the adult. This _seems to be _a natural consequence of mutation accumulation. We intend to indude in _a future

study

trie effects of _a

slowly evolving

environment. We will allow the

"optimal"

_sequence to

slowly

mutate with time. This will introduce aspects ^{of the}

Bak-Sneppen

_[Si model where co-evolution

of _a

self-organised

critical system ^leads to extinction _on all scales. Another

topic

^of interest is what

happens

when _a

parental

_care factor _is introduced into the model.

Any youth

with _a

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Ave~ge Mu~tion w& Age

D.Q

° o

0.8

~ ~ o

o

à Ù-fi o

fl S o

f

_0.5

ce °

fi

g o

< °.~

o 0.3

o

0.2 Q

o

°

0.1

~ ~ o ° ° ~

~ o o o o °

~ o ° °

0

0 5 10 15 20 25 30 35

AGE

Fig. 2. The average number of mutations per individual for each age group with the fertility _range being from Ii to 25. Note trie rapta increase at 15 years.

AGE ATDEATH

o

o ~

6000 ° °

o

o ~

5000 °

o o

~

~ o

o o ° °

° ° o o ~

4000 z

" o

à

j~ ³⁰⁰⁰ ^°

o

2000

o

1000 o

o

° o o

~

o 5 10 15 20 25 30 35 40

AGE

Fig. 3. The frequeucy of death with _age for the _case where the fertility _range _is from Ii to 25. The first peak is sensitive to number of hereditary point ^mutations ^allowed at birth.

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Tata Mutations withAge 35

o o

30 ^°

o

~ o

25 °

o o

°

o ~

fl 20 ^° °

o

O °

P ^°

~ o ~

1

₁₅

° o

o io

o

5 _o

o o

o ^°

0 5 10 15 20 25 30 35

AGE

Fig. 4. The _sum of the mutations for the individuals _in each age group, independent of the _size of each group.

Sumkal Ratio wth Age

o o

o o-fi

S ^~ _o o

4 o ~ o

% É

z o o

0.4

~ o

0.2

0

0 5 10 15 20 25 30 35

AGE

Fig. 5. The survival ratio _as _a function of age for the _case where the fertility _range is from ii to 25.

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living

parent ^would ^have a greater chance of

surviving.

Acknowledgments

We welcome the opportuuity to thank Professor Stauffer

(who

is

performiug self-experimeuts

ou

ageing)

and his collaborators

Penna,

Moss de Oliveria and Bemardes for

keepiug

_us informed of their work aud also for encouragement. ^This ^research ^is

supported

in part

by

NSERC of

Canada and _a UCR grant ^from ^St. ^Francis ^Xavier

University.

References

Ill

Rose M.R., Evolution Biology of Aging

(Oxford

University Press, Oxford, 1991), and _see refer-

ences therein for earlier literature; Charlesworth B., Evolution

in Age-Structured Populations, 2nd edition

(Cambridge

University Press, Cambridge,

1994).

[2] ^Penna T.J.P., J. Star. Phys. 78

(1995)

1629; Bernardes A.T. and Staulfer D., submitted

(1995);

Penna T.J. and Staulfer D., Înt. J. Med. Phys. ^C6

(1995)

233; Moss de Oliveria S., Penna T.J.P.

and Staulfer D., Physica A215

(1995)

298. Earlier simulations _are reviewed by Staulfer D., Braz.

J. Phys. 26

(1994)

900.

[3] Amitrano Ç., Peliti L. and Saber M., ^Molecular ^Evolution _on Rugged Landscapes: Proteins, ANA

and the Immune System, A.S. Perelson and S-A- Kaulfman, Eds.

(Addison-Wesley

Publishing Co.,

Redwood City,

1991).

[4] Ridley M., Trie Red Queen

(Macmillan

Publishing Co., New York,

1993).

[5] Bak P. and Sneppen K., Phys. ^Reu. Lent. 71

(1993)

4083; Ray T.S, and Jan N., Phys. Reu. Lent.

72

(1994)

4045.