Exact Results of the Bit-String Model for Catastrophic Senescence

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Exact Results of the Bit-String Model for Catastrophic Senescence

T. Penna, S. Moss de Oliveira

To cite this version:

T. Penna, S. Moss de Oliveira. Exact Results of the Bit-String Model for Catastrophic Senes- cence. Journal de Physique I, EDP Sciences, 1995, 5 (12), pp.1697-1703. �10.1051/jp1:1995225�.

�jpa-00247169�

J.

Phys.

I France 5

(1995)

1697-1703 DECEMBER1995, PAGE 169î

Classification

Physics

Abstracts

05.50+q 89.60+x 07.05Tp

Exact Results of trie Bit-String Model for Catastrophic

Senescence

T.J.P.

Penna(~)

and S. à~oss de

Oliveira(~*)

Instituto de Fisica, Universidade Federal Fluminense, Av. Litorânea

s/n,

Boa

Viagem,

24210-340

Niterôi, RJ,

Brazil

(Received

14

August

1995,

accepted

in final form 18

August 1995)

Abstract. We succeeded in

obtaining

exact results of trie

bit-string

mortel of

biological aging

for

populations

whose individuals breed

only

_once. These results

ai-e in excellent agreement with those obtained

through

computer simulations. In addition, _we obtaiu _an expression for trie

minimum birth needed _to avoid mutational meltdown.

1. Introduction

The

aging

is characterized

by

trie loss of

capabilities;

in _an irreversible _way,

predominantly

after trie

maturation,

for trie most of

species. Mary

factors _can influence trie

aging

_process:

biochemical _processes, environment and _so fortin.

Recently,

_many efforts bave been directed towards _an

evolutionary theory

^of

aging [1-3].

Trie fundamental

question: Why people get

old ? has been

recently replaced by: ivhy

is the survival

probability

of _an individual reduced

with

advancing

_age ? The _reason is the latter is _more suited to be answered with the tools of

dynamics

of

population

and Mathematical

Physics methods,

_as

suggested by

Rose

iii.

For

a review of Monte Carlo simulations _in

biological

_{aging see,} for instance, reference

[4].

^The

theories of

evolutionary

aspects of _aging fall in two classes:

optimality

the

optimal

life

strategy

is _an

emphasis

_on

early

maturation

tl~at,

due to

pl~ysiological

constraints. leads to _a decrease in tl~e life _span; and

genetic explanations

deleterious

(bad)

mutations for old _age

are

subject

to a weaker selection tl~at tends to accumulate tl~em later _in tl~e life time.

Sometimes,

_aging effects _are

displayed

in _an amazing way, like in tl~e so-called

catastropl~ic

senescence. It is _a well known fact tl~at

semelparous individuals,

1-e-, those wl~icl~ breed

only

once,

usually

_present

catastropl~ic

senescence, that

is, tl~ey

aie

sl~ortly

after

reproduction _[3].

Owing

to its commercial

i>alue,

tl~e Pacific salmon _{is one} of tl~e most

important exaiuples.

The

catastrophic

_senescence has been studied in _a mortel of

optiniality (Partridge-Barton mortel) [3, si. Nevertheless,

if _we

accept

^{that natural selection}

pre,>ents

^tl~e manifestation of deleterious

(" ^e-mail:

tjpp©if.uff.br

(** ^e-mail: suzana@if.uff. br

Q

Les Editions de

Physique

1995

1698 JOURNAL DE

PHYSIQUE

I N°12

mutations before

procreation,

it is easy ^to understand

why

tl~is

premature

senescence occurs:

after

reproduction,

tl~e individuum does not need to be

protected

_anymore, since it is not

going

to

produce offspring

_any

longer.

Tl~e

bit-string

mortel of life

l~istory, recently

introduced

[6, 7], specifically

makes _use of _a bal-

ance between

l~ereditary

mutations and

evolutionary

selection pressure.

Computer

simulations of tl~e

bit-string

mortel bave sl~own tl~at the

catastropl~ic

_{senescence can} also be understood

~vitl~in the framework of mutation accumulation

theory. Furtl~ermore,

it _was

possible

to show

that _{senescence is a} direct result of tl~e age of

reproduction

_[8].

Here,

we present

analytical

calculations whicl~ confirm that tl~e life

strategy

^of

reproducing only

_once leads to tl~e catas-

trophic

senescence. This _{paper is}

organized

_as follows: in Section

2,

_we

briefly

describe the

bit-string

mortel. In Section

3,

we derive _our

analytical

results and discuss them

by

_companson

witl~

computer

simulations.

Finally,

_m Section 4 _we

present

ouf conclusions.

2. The Model

Consider _an initial

population N(t

=

0)

of individuals eacl~ _one cl~aracterized

by

_a word of B bits. Tl~is word is related to tl~e

genetic

code of eacl~

individual,

since it contains tl~e

information of iv-hen tl~e effect of _a deletenous mutation will take

place (bit

set to

one)

or not

(bit

set to

zero) dunng

the life of the individual. We _can consider the time _{as a} discrete variable

running

from to B years.

Hence,

if at time t _{= i} the i~~ bit in tl~e word

equals

_one, it is considered tl~at tl~e individual will suffer tl~e effects of _a deleterious mutation

in that _year and all

following

_years. The mutations _are

hereditary

_in tl~e _sense

tl~at,

at tl~e birth of _{a new}

individual, they

_are transmitted to tl~is _new individual except for M bits

(M

will be called mutation rate,

hereafter), randomiy

chosen. These mutations _con

correspond

to either _an additional deleterious mutation _{or a}

cleaning

of _a deleterious mutation

present

at the

parent's

_genome. We _can interpret ^this new mutation _{as a} point ^mutation

(if

_{we are}

considenng

asexual

reproduction)

_{or a} recombination of the

parent's

_genome in sexual

reproduction.

An individual will aie due to aging effects if the number of deleterious mutations tl~at _were relevant up to its age is

equal

to _a tl~resl~old T.

Competition

betiv-een tl~e individuals of _a

population

for food and space, for

example

are introduced

taking

into accourt an

age-independent

Verhulst

factor,

wl~icl~

gives

to eacl~

individual _a

probability

_{x =}

il Ntot(t)/Nmax)

of

staying

alive.

Nmax

_means tl~e _maximum size of tl~e

population.

Eacl~

surviving

individual

generates,

at tl~e

reproduction

_age R, b

offspnngs.

As stressed in reference

[9],

^{tl~e fact of} individuals breed

only

_{once in} their lifetimes

is tl~e fundamental

ingredient

to

explain

tl~e

catastropl~ic

senescence.

3.

Analytical

Calculations and Results

The relevant parameters ⁱⁿ our calculations _are:

. B

genotype lengtl~,

_in

bits;

. b birth

rate;

. R the

reproduction

_age;

. x the Verhulst

factor;

. iv number of deletenous mutations added at birth.

. T tl~resl~old

starting

from ~vl~ich tl~e mutations lead to deatl~.

N°12 BIT-STRING MODEL FOR CATASTROPHIC SENESCENCE 1699

Let _{us assume} tl~at tl~e system bas reacl~ed _an

equilibrium

state

(tl~is liypotl~esis

is

plausible

since

computer

simulations bave sl~own tl~at it al1&,ays

occurs).

Since tl~e age distribution and

tl~e

population

are

stationary,

_{we can} consider tl~e Verl~ulst factor _as constant. In tl~is

work,

for tl~e sake of

simplicity,

_{we assume}

only

deleterious mutations at birtl~ witl~ T

=

1,

_1-e-, tl~e effects of

just

_one deleterious mutation is

enougl~

to _cause tl~e deatl~ of _an individual. Tl~e extension for different values of T is

straigl~tforward, altl~ougl~

_we bave learned from

previous

computer

simulations results tl~at it is trot

important

for tl~e

catastropl~ic

senescence

exploration.

Let _us consider _one bas Jh~ individuals with _{zero age,} after trie

stationary

state has been reacl~ed. The

probability

of _a deleterious mutation _to appear ^at a

given

_{age is}

(tif/B), for

_ages

beiow

R,

since _in this _{case we con} be _sure that the

parent

^bas no mutations up ^to age R

(T

=

1)

and tl~e mutations at birtl~ _are

randomly

cl~osen

(we

_are

considering

tl~e _case AI _«

B).

For ages

beyond R,

we must take into account the whole

parent's l~istory,

and tl~is will be clone later. First

ignoring birtl~s,

_{we con}

expect

^tl~at tl~e number of individuals witl~ age 1 will be

Ni =fif~ Ii j) _=fifx

~~ j~). ₍₁₎

Analogously,

~~ ~~~ ~

B~11 ^{~~~ ~~ B~~ ~(B~~l)}

^{~~ ~~~}

is tl~e number of individuals witl~ age 2. For

general

_age k <

R,

_one bas

~~ ~~~ ~~

_~

B~- i~ÎÎ

_~~~

Now,

_we take into account tl~e

reproduction

of trie individuals _{at age} R. Since eacl~ individuum

produces

b

offsprings,

tl~e number of individuals witl~ _zero age will be b times tl~e number of age R in tl~e previous step:

~~

~

~~ ~~

_~

B~- iÎÎ

^~

~~~

Using

tl~e above

equation,

witl~

tif

=

No,

_{one can} obtain tl~e

stationary

state condition

x~ fl

^{~ ~ ~}

= i

(5)

~i

~~~+~

As _we sl~ould expect, tl~is condition does not

depend

_on k but

only

_on tl~e

global

_parameters

R, M,

b and B. For M

= 1 it reduces to

bx~(B R)/B

= 1.

For ages k >

R,

we must

garantee

tl~at _no mutation occurred between ages R and k. The

probability

that _no mutation has occurred until age

k,

in the t~~

=

1,

^2. ancestor

(parent,

grand

parent,

etc.)

_is

(see Eq. (3)):

lk

^{~ ~}

§ _B~-1+Î~~'

~ _~~~

and this _{is a} necessary condition for the individual to be still

olive,

to be fulfilled for all ancestors.

However, except

^{for M} ₌

0,

this term goes ^ta zero as t increases. In tl~is 1&>ay, in order to bave tl~e

stationarity

condition

satisfied,

_we must bave

Nk>R

" 0.

(7)

lî00 JOURNAL DE

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I N°12

o i

S

N°12 BIT-STRING MODEL FOR CATASTROPHIC SENESCENCE 1701

ioo

'Éc

~ io

u

0 2 3 4 5

M

Fig.

2. Minimum birth rate bm~,, us. mutation rate IV. Trie solid fines

correspond

to

analytical

results for R

= 5,10, là, 20 from trie bottom to trie top. ^Trie

symbols correspond

_{to computer sim-}

ulations R

=

5(D),10(O),15(li)

and

20(o).

In trie computer simulations, _{we cari}

only

obtain trie

maximum integer value below which trie

population

vanishes

(mutational meltdown).

That is

whj,

these results _are belo~v the theoretical results. trie do not present results for

larger

values of b because it would _{reqmre a}

huge

number of time steps until trie population vanishes.

that tl~e mutational meltdowii does Rot

depend

_on trie size of

population

at least for asexual

semelparous

_species in contrast witl~ earlier

biological assumptions

tl~at tl~is effect _occurs

only

_on small

populations.

~Îe present in

Figure

2 _{some curves} for

b,n,n

_{as a} function of Jf.

For

comparison,

_we

present

some results from computer simulations. However, siiice _we bave 1&>orked

only

with

integer

values of

b,

tl~e tl~eoretical results should be considered _{as an} upper limit. For

large

values of b

(typically

>

10)

_{and l~igl~} mutation rates

(fil

_>

3),

tl~e number of time

steps

needed to reacl~ tl~e

equilibrium

is very

large.

For _{a more}

precise

determination of

bm,n,

_we tried _an

exponential fit,

for

Ntot

_us. t

(for

10000 _< t <

100000),

and _we cl~ecked trie

sign

of tl~e exportent to determine wl~ether the

population

would

eventually

vaiiisl~. Since tl~e

study

of

dynamical _aspects

of tl~is mortel _are Dot into the scope of this work, _we present

only

the _more reliable results.

Trie relevant

quantities

in aging studies _are tl~e survival _rates

Sk,

_{1-e-, trie}

probabilities

^of _an

individual to survive to the next age.

Tl~ey

_cari be

easily

obtained front

equation (8):

~~ ÎÎÎI

^~

_B~ kÎÎ ^Î

B

~i ~MÎ1~

^~~~

~~~~

Tl~e survival rate for age k

gives

tl~e

probability

that individuals alive at age k -1 survive

and reacl~ age k. In

Figure 3,

we compare tl~is expression for different values of b and R1&~itli computer ^{simulation results. Tl~e}

abrupt jump

to _zero at k

= R is

clearly

_seen.

lî02 JOURNAL DE

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.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-._._.

0.8 ~~~~~~~~~~

,

j~ce

~ Off lC

)

(

_0.4

Ù7

o_2

°'°~

~ 10 lS 20 25

Age

Fig.

3. Survival rates _us. age. Here, _we present ^{trie results for M} = 1,b

= 5, R

= II

(O

and sohd

bne),

R

= 20

(.

and dashed

hne)

and b

= 10, R

= II

(li

and dotted

fine).

We followed trie _same strategy of averagmg as

adopted

in

Figure

1.

Again,

trie agreement between

theory

and computer

experiment ^is renlarkable.

4. Conclusions

Using simple

statistical arguments, we bave solved

analytically

tl~e

bit-string mortel,

for semel- parous

species

as tl~e Pacific salmon and

mayflies.

Dur results confirm tl~ose

presented

in reference

[9],

^{where it is claimed tl~at trie life}

strategy

of

just

_one

breeding attempt

leads to

catastl~rophic

senescence,

according

to tl~e mutation accumulation

tl~eory.

In

addition,

_{we can} obtain _an

analytical

_expression for the

population

_size of

semelparous

_{species as} function of

mutation rate and

reproduction

_age

(an exponential decay

for the _same conditions of _space and

food),

and tl~e minimum birtl~ _rate to avoid mutational meltdown. Dur results also agree witl~ tl~ose found

by

Bernardes and

Stauffer, suggesting

tl~at trie mutational meltdown is not an effect of small

populations. Nevertl~eless,

we bave

found, by

_computer

experiments,

tl~at tl~e time

required

for _a

population

to vanisl~ _con be very

large.

In this sense, we

suggest

tl~at it _can be wortl~

studying

tl~e

dynamical aspects

^{of tl~e}

bit-string mortel,

for botl~

semelparous

and

iteroparous

_species.

As _a final remark _we wish to

point

out tl~at

altl~ough dealing

witl~

semelparous organisms,

tl~e extension for

iteroparous organisms (whicl~

breed

repeatedly)

is

straightforward,

but the

corresponding expressions

are too

lengtl~y

to be useful.

Acknowledgments

This work _was motivated

by

_a

question

raised

by

Naeem Jan about _a

previous

_paper. ^We

are

grateful

to hiin for bis interest in _ouf work. Due of _us

(TJPP)

_is also

grateful

to Silvio Salinas and Nestor Caticl~a for

encouragement,

^{and tl~e otl~er}

(SMO)

to P-M-C- de Oliveira for

important suggestions,

and to Hans J. Herrmann for bis

l~ospitality

at ESPCI wl~ere tl~is

work _1&,as concluded. Tl~is work is

partially supported by

Brazilian

agencies CNPq, CAPES,

FAPERJ and FINEP.

N°12 BIT-STRING MODEL FOR CATASTROPHIC SENESCENCE 1703

References

iii

Rose M.R., Trie

Evolutionary Biology

of

Aging (Oxford

Univ.

Press,

Oxford

1991).

[2] Charlesworth B., Evolution in

Age-structured

Populations, 2nd edition

(Cambridge

Univ. Press~

Cambridge 1994).

[3]

Partridge

L. and Barton

N-H-,

Nature 362

(1993)

305.

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D.,

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N.,

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T.J.P.,

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