Ageing: Parental Care and Reproduction Strategies

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Ageing: Parental Care and Reproduction Strategies

John Thoms, Peter Donahue, Doug Hunter, Naeem Jan

To cite this version:

John Thoms, Peter Donahue, Doug Hunter, Naeem Jan. Ageing: Parental Care and Reproduction Strategies. Journal de Physique I, EDP Sciences, 1995, 5 (12), pp.1689-1695. �10.1051/jp1:1995224�.

�jpa-00247168�

Classification

Physics

,4bstracts

02.50-1 05.20-y 87.10+e

Ageing: Parental Care and Reproduction Strategies

John

Tl~oms,

Peter

Donal~ue, lJoug

Hunter and Naeem Jan

Physics Department,

St. Fian< _is Xavier University,

Antigomsh,

Box 5000, Nova Scotia,

B2G 2W5 Canada

(Receivedls August

199?à, ^received in final form and

accepted

18

August 1995)

Abstract. Trie Penna

bitstring

mortel of

biological ageing

bas been

successfully

used ta illustrate _senescence due to mutation accumulation. We introduce several _new factors includ- mg parental _care which

produces

_a noticeable increase in trie fitness and life expectancy ^{of trie}

population,

_a

changing reproductive

_range which shows trie distinct

relationship

between _senes-

cence and

fertility,

and

semelparous

mdividuals which

produce

several _new observations such

as a

cyclic population.

1. Introduction

Altl~ough

there _are niany diiferent types ^of

animais,

trie

general

pattern ^{of their lives} is trie _sanie.

ivitl~ tl~e

completion

of

puberty,

_we

experience

increased

pl~ysical

detenoration.

However,

tl~e rate of tl~is deterioration diifers for eacl~

species. Pl~ysical

detenoration witl~

time,

rather tl~an

siniply growing aider,

is wl~at is nieant

by

the term

ageing.

Yet the

'Why?'

and tl~e 'Howl?' of

ageing

remains somewhat of _a

mystery.

However there _{is growing} evidence which links

senescence

(or ageing)

to the

theory

of _mutation accumulation and

reproduction

behaviour

iii.

This

theory

asserts that _ageing is the result of

fate-acting

deleterious mutations which _ai-e

weakly

selected

by

raturai selection at

reproduction.

The _presence of these 'bad' mutations in the individual _are felt fate _in lire and decrease the

ability

of the

organism

to carry ^ont its normal

functions;

thus

making

the

organism

more

susceptible

ta

injury,

disease, and then death.

Having explored

_aspects of this

theory

_{[3], we now}

study

_more

closeljr

the

relationship

between

ageing

^and

reproduction

behaviour:

namely semelparous individuals,

which breed

only

_once,

iteroparous individuals,

which breed

repeatedly, genetic

inheritance of

reproduction

information,

and the

reproduction

effort of

parental

_cane.

2. General Model

We

give

_a brief

description

of the mortel of mutation accumulation used in _Dur

study.

A modified Penna

I?i 'bit-string'

mortel is used. The mortel has either 3? _or 64 genes

representing

an

individual,

which may be either 'On' _or 'Off'

il

_or

0).

At the start _we

generate

a random

g

Les Editions de

Physique

1995

1690 JOURNAL DE

PHYSIQUE

I N°12

sequence of states to represent

optimal

fitness for the environment. Given _a

population

_we

increment the _{age, 1,} of each individual in the

population by (a year)

and this is

represented by

the

expression

of the i~~ gene. The

newly expressed

_gene is

compared

to its

counterpart

in the

optimal

_sequence. If

they

_agree, then this represents a

'good'

_gene; if trot, then this

is considered _a deleterious

hereditary

mutation. The

Haniming

distance _or total deleterious niutations between the active

segnient

and the

corresponding segment

of the

optimal

_sequence is used to compute a death

probability.

We also allow at each time

step

^which we refer _{to as a year,} witlt _a

probability

of

0.01,

for each individual _to

acquire

_a deleterious somatic mutation. Note that this _unit of time is

species dependent

and

represents

a convenient interval

representing

an observable

change

_in the

organism.

A somatic mutation is _one that is detriniental to the

organism,

but is trot

passed

_on to the

ofispring.

A Fermi function

involving

the average number of mutations b

(hereditary plus somatic),

the actual number of mutations of the individual concemed _a, and _T, which

plays

the rote of

an inverse

temperature,

is used [3] ^to

compute

a death

probability

_pdi

~~

exp(T(/ ^a))

₊ ^~~~

This death

probability

is then

compared

with _a random number

ranging

from _zero to one.

If the random number faits to exceed the death

probability

then the mdividual _is removed from the

population.

An

mdependent

term the Verhulst Factor

il N(t)/K)

where K is the

carrying capacity

of the environment and

N(t)

is the

population

at year ^t functions _m

a similar _manner

removing

individuals when _a second random number exceeds the Verhulst term.

Ne~v organisms ^are introduced into the

population

each

having

the identical

genetic string

of its

parent

^but we allow with

equal probability

for either

0, 1,

or 2

hereditary point

mutations at birth. This is clone

by simply switching

_a

randomly

selected _gene of the

parent

^from its

present state to the

opposite

state.

3.

Ageing

aud

Reproduction

Behaviour

3.1. REPRODUCTION RANGE _AND AGEING IN AN ÎTEROPAROUS POPULATION. TO SÎIOW

clearly

the

relationship

between the mutation accumulation

theory

of

ageing

^and

reproduction

range we present ^the results of two

separate

ruas in which ail

parameters except

the

reproduc-

tion range are

kept

constant. For both sets of data the

population

is monitored to _ensure that

we have reached the

steady

state. These results _are for _a

population

whose members maxi-

mum life _span is 64 years.

Figure

shows the

steady

state age distribution for the

reproductive

ranges of10-25 and 20-35. Note how the _curve

representing

the older

reproductive period

has

an extended _range when

compared

_to the

former,

with the demise of the

population

_{now com-}

ing

at the end of the _new

fertility

_range,

35,

rather thon _at 25. The correlation between _ageing

and the

fertility

of _an individual is

dearly

demonstrated.

As the

reproductive

_range is shifted

upward

for the _same initial conditions the size of the

steady

state

population

decreases.

Starting

with _an initial

population of10000,

with half

being

identical to the

optimal

_{sequence we} find for the

reproductive

_range of10-25 the

population

_size

is

just

over 8000. For that of14-29 it is under 6500 and for the

reproductive

_range of 20-35 the

population

_size is close to 4200. We observe that the average

steady

state

population

decreases almost

linearly

with increase in the minimal _âge of

reproduction.

Also _as the

reproductive

range is shifted

upward

the fitness for _any

particular

_{age group} also increases. For instance the average number of mutations for the age group of13-25 for the

reproductive

_range of 11-25 is 1.65 _as

compared

to 0.23 for individuals _in the _same age group where the

reproductive

_range

iooo

goo

Hep Range 10.25 ~

~

Hep Range20.35 +

800 _~

700 ^~

~ ₆₀₀ ~

o ~

~

é 500 ^~

É

Î ~

fl 400 ~

~

~ +

~ ₊ o

~ ~

+

~ ~

zoo ~

+

+ +

~ + o

+ + o

~ + +

+ Î

q

~ loo

~

~ ~ ~ +

+ + +

~

o 5 1° ~

ÀÎe ^{~~ ~~ ~~ ~~}

Fig.

l. Trie age distribution of trie

iteroparous population

where trie

reproductive

_range is 10-25, and where it is 20-35. Ail other parameters _are

kept

constant. Note that _as trie

reproductive

_range is shifted

upward

the _size of the

steady

state

population

is

reduced,

while the fitness of the

population

mcreases.

of 20-35 _an

improvement

of _over

80%.

A

plot

ofthe fitness in the _range 13-25 shows _a

sharp

decrease with

increasing

shift in the

reproductive period, reaching

_a minimum for values of the

reproductive period

from 16-31 and

greater.

^The average number of mutations with age and tl~e _age at wl~icl~ deatl~ _occurs show similar

patterns

for tl~e different

reproductive

_ranges but

are soniewl~at stretcl~ed _to include tl~e end of tl~e

fertility

_range.

3.2. SEMELPAROUS REPRODUCTION.

Looking

at trie nature of

senlelparous reproduction

habits,

_we consider _a mortel wl~ere each member of the initial

population,

whose maximum life _{span is} 32 years, is

given

a

single

_age

(ranging

from

10-32)

at wl~ich

they

would

give

birtl~

to 0-3

offspnng

with

equal probability.

This âge ^at which

reproduction

_occurs is then

passed

on to their

offspring.

As _{seen in}

Figure 2a,

within 1000 years ail members of the

population

with

reproductive

_ages other thon 10 l~ad died out,

leaving

_a

population

of individuals wl~o

reproduce simultaneously

_every ter years, witl~ _{a aine} year gap when _{no new} individuals _are boni

(as

sl~own in

Fig. 2b). Naturally

this led to _a 10 year

population cycle

wl~icl~ is restricted

by

tl~e Verl~ulst factor.

This

cycle migl~t

hâve been initiated

by

tl~e fact tl~at _our initial

population

ail had _{an âge} of 0 and tl~us ail tl~e

people

will

reproduce

at ten years of _age and tl~eir

offspring

will

give

birth at year

20,

and _{so on.} To _overcome this _we

assign

_a random age ^to each member of the initial

population.

Tl~e mdividuals who

reproduce

at ages other tl~an ten died out _once

again,

but this time there _are several groups of

people giving

birth in each year of the ten year

quasi periodic cycle. (ie.

One _group

giving

birth at

10, 20, 30...,

anotl~er

giving

birth at

Il, 21,

31..

and _so

on.)

As time went on

l~owever,

ail tl~e _groups died out

except

^for _one, as seen in tl~e birtl~s witl~ time

grapl~

in

Figure

3. Tl~is _group exl~ibits tl~e _saine ten year

cycle

_as tl~e previous

run of this

type (Fig. 2b).

The

phenomenon displayed

in

Figure

3 where _{one group}

eventually

dominates the

population

is very similar _to that found in the modified Ehrenfest _um mortel _[4]

in which _an

equal

number of10 different coloured balls _are

placed

into _{an urn.} Half of the balls

1692 JOURNAL DE

PHYSIQUE

I N°12

20

15

w1

(

w

é ₁₀

9

~

oe

o

o zoo 400 600 800 looo

Year

&)

loooo

,Number OfBinhf

, j ^, PoiulationwithTimfl

1, .; il ') Il

ÎÎ 'Î )(

lÎ Î1

>' 1.

'>

8000

;

~~ ~ ~~

~ ~

"

".

~

"

2000

o

15000 15010 15020 15030 15040 15050 15060

Year

b)

Fig. 2. Trie _{average age} of reproduction with time for

a semelparous population. Age of reproduc- tion for trie initial

population

_is

randomly

set to be

a

surgie

_{year m} trie range la 32. A combination of trie number of births _{per year} and trie

population

_size with time

(in years),

for trie time

period

of

15000 to 15060.

are tlien removed followed

by doubling

the number of balls of each colour _in trie _um. This

procedure

is then

repeated by

_removing ^half the balls and

continuing

^the _process. ^With time

only

_one colour of the

original

10 _remains.

Having

a

rigid siitgle

_year of

reproduction

would al1&>ays lead to the type of

population

and birth

cycle

(lescnbed above. Therefore _we

give

each individual _a

single

_age at which to

reproduce,

but hm,e this _âge chosen

randomly

to be

10,

ii, or

12,

for both the initial

population

and their

offspnng.

This gai>e a much _more uniform

population

_over time and dia

7000

g

à 4000

à

3

~

fl 3000

~

=

2000 ''

'.>.~ f"'

~

i wJ'Îj 'Î "

1000 j j' i

w'.

~

'" ^~

"~.' ^~4) '

', ~.' [ ~"N~

~

o 2000 4000 6000 8000 loooo 12000 14000 16000 18000 20000

Year

Fig.

3. Trie number of births _{per year} recorded for 20000 years. Each fine represents a different reproductive _group.

(See text)

net exhibit the 10 year birth

cycle

frein _our earlier _runs.

Catastrophic

_senescence is observed if the value of _T in the Fermi

type

death

probability

is increased. _T

plays

the rote of the _inverse

temperature

^which

corresponds

to trie tolerance of the environment to1&>ards differences of _an mdividual from the _average fitness. We found that

by increasing

the value of _T

thereby decreasing

the tolerance of the environment from 10 to 100 extreme senescence is observed after

reproduction.

This result confirms the main conclusion of Penna et ai.

[si.

We consider another variance of the mortel where each individual is

given

_a

single

_âge at which

they

would

give

birth

(10, ii,

or

12).

At this time ho1&>ever,

90%

of the children _were given tl~e age of

reproduction

_as tl~eir

parent

witl~ tl~e

remaining ^10%

_given a random âge of

reproduction.

Tl~e

resulting steady

state

population

bas _an average âge of

reproduction

of 10.56 and _{an average age} of deatl~ of 12.21. In tl~e

steady

state, births teck

place

_{every year,}

as

opposed

to ail

being

in tl~e _same year, so the

population

dia not follow _a

longterm penodic cycle.

~vl~en tl~is _{rua is}

repeated

^{witl~ cl~ildren}

ha,>mg

_a

99%

chance of

reproducing

at tl~e

same age as tl~eir parent, ^the _average ^of age of

reproduction drops

to 10.16 and the a,>erage age of death faits to 11.67.

Agoni

the

population

showed _no

longterm periodic cycle,

and not ail tl~e births occured in the _same year. Tl~e

greater

^{trie chance of}

having

_a birtl~ at trie _same age as their

parent

trie

stronger

tl~e

tendency

tov>ard _senescence after trie age of

reproduction.

3.3. PARENTAL CARE _{IN AN} ITEROPAROUS POPULATION. Trie above results _ai-e observed

for mortels wl~icl~ dia net take into account

reproduction

effort.

By reproduction

effort _{we mean} tl~e energy devoted to secunng mates, to

reproducing

wl~ich is

dependent

_on the prospects for future

reproduction

-, and investment in tl~e

offsprmg

after birtl~

(parental care).

To observe tl~is effect tl~e

bitstring

mortel is

adapted

to inclttde tl~e

concept

^of

parental

_care.

This _is tl~e first time to _ouf

knowledge

that anyone lias studied the effects of

parental

_care.

Parental _{care is} introduced into trie mortel trot 1&:ith trie parents receiving any

davantage,

but with their

offspnng doing

so. Trie mdividual whose parent is still olive evades trie

population

size check of the Verhulst factor.

However,

this

places

older members of the

population

at

1694 JOURNAL DE

PHYSIQUE

I N°12

1600

NO Parental Care o

+ With Parental Care +

1400 ⁺

+

+

+

© +

i~

~ ° +

É

Î

~ o

Z ~

o

+ o

o o

+

o ⁺

o ~

o

~

°

o ⁺

°

o ~

o $ +

° o

0 5

Fig.

4. Trie age distribution of trie iteroparous

population

with and without trie

reproduction

effort of

parental

_care. Note that trie introduction of

parental

_care increases both the

size of the steady state

population

and the bfe expectancy.

a

disad,>antage

since

they

will be _{now more}

likely

eliminated

by

the Verhulst

factor; _yet

the

major

result observed from the introduction of

parental

_care is the noticeable increase in life

expectancy

^{and in} the overall fitness of the

population.

Note the difference between the _age distribution

graphs

in

Figure 4,

without

parental

_cane, and of the case

incorporating parental

care. The average number of mutations for the age group of13-25 without

parental

_{care [3]}

is

I.Il, whereas,

_m the trial

including parental

_care it is

0.48;

_a

70% improvement.

For the

run 1&,ith

parental

_care the âge ^at death _curve is less dramatic than trie _run without

parental

care, with the second

peak being

more of _a

bump

followed

by

_a

sharp

decline at the end of the

fertility

_range. This data _is collected from trie

study

where each individual has _{a maximum}

lifespan

of 64 and _a

reproduction

_range of10 25. For _an initial

population of10000,

the

steady

state iteroparous

populations

for the two _cases differ

by 6000,

with the

steady

state

population being

^{15000 with}

parental

_care and 9000 without

parental

_care.

4. Conclusion

We have shown that the modified Penna

bitstring

mortel for mutation accumulation

theory

of

ageing

is well suited to sholv- the

relationship

between _senescence and

reproductive

behavior.

The

relationship

between

fertility

and ageing ^is

clearly displayed,

with the demise of the _popu- lation commg ^at the end of the

reproductive

_range.

Also,

_we bave illustrated the trend toward

population cycles

for

semelparous

_species with

ngidly

mhented _ages of

reproduction,

and the overcommg of this trend when the age of

reproduction

is allowed to vary

slightly.

With the introduction of

parental

_cane into the

mortel,

_we observe _{an increase} in the life expectancy ^and overall fitness of the mdividuals in the

population.

^Future areas of

study

_on the mechanisms

of _agemg and

reproduction

_may mdude the introduction of _a

changing

environment, and _a

population

which

reproduces sexually.

Acknowledgments

We welcome the

opportunity

to thank Professor Stauffer and his associates at KoIn

University,

as well _as Professor Penna and his collaborators at Universidade Federal

Fluminense,

for

keep-

mg us informed of their work and also for their

encouragement.

^We are

obligated

to Rachel Bland for her comments and

support.

This research is

supported

in

part by

NSERC of Canada and _a UCR

grant

^{from St. Francis Xavier}

University.

References

iii

Rose

M.R.,

Evolution

Biology

of

Aging, (Oxford

Univ. Press Oxford

1991)

and _see references

therem for earlier hterature; Charlesworth B., Evolution _m

Age-structured Populations,

2nd

edition,

(Cambridge

Univ.

Press, Cambridge 1994);

Stauffer D., Braz. J.

Phys.

24

(1994)

900.

[2] ^Penna

T.J.P.,

J. Stat.

Phys.

78

(1995)

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

(1995);

Penna T.J.P. and Stauffer D., Int. J. Med.

Phys.

C 6

(1995)

233; Moss de Oliveira

S.,

Penna T.J.P. and Stauffer

D., Physica

A215

(1995)

298; Vollmar S. and

Dasgupta S.,

J.

Phys.

I France 4

(1994)

817.

[3] ^Thoms J., Donahue P., Jan N., J.

Phys.

I France 5

(1995)

935.

[4] ^{Johnson N-L- and} Kotz

S.,

Urn Models and their

Applications:

An

Approach

to Modem Discrete

Probability, (Wiley,

New York

1977).

[5] ^Penna T.J.P., Moss de Oliveira S. and Stauffer D., to appear m Rapid Communications _m

Phys.

Reu..