A model for ageing with hereditary mutations

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A model for ageing with hereditary mutations

Stefan Vollmar, Subinay Dasgupta

To cite this version:

Stefan Vollmar, Subinay Dasgupta. A model for ageing with hereditary mutations. Journal de

Physique I, EDP Sciences, 1994, 4 (6), pp.817-822. �10.1051/jp1:1994228�. �jpa-00246948�

Classification Physics Abstracts

05.40 87.10

Short Communication

A mortel for ageing with hereditary mutations

Stefan Vollmar and

Subinay Dasgupta (~,*)

Institute of Theoretical Physics, Cologne University, 50937 KüIn, Germany

(Received

13 April 1994, accepted 19 Apnl

1994)

Abstract. We show that _a simple model that incorporates _a balance of helpful and delete-

nous mutations inherited by future generations, ^leads to _an 'explanation' of biological ageing.

We aise investigate trie stability hmit beyond which the populations decay to _zero.

For ages,

biologists

and

ecologists

have been trying to

explain why living beings

_age and

why

the

populations

of _{some species} show

interesting temporal

behaviour

je-g-,

remain

constant)

[1].

Recently,

it has been found that in this connection it is aise very

helpful

to take _recourse to the

techniques

of

computational physics, mainly

those suited for systems made of _a

huge

number of

interacting

elements. In _one of such

works,

Staulfer and Jan [2] have shown that

a

simple

'discrete' model

incorporating

somatic mutations _can

reproduce

_some

ageing

elfects.

Ray

_[3] continued this

approach by investigating

the

dynamics.

The present communication aims _at certain

investigations

related to this work. We present

some results which _can be the

subject

of future research. In _a

nutshell,

the work

by

Staulfer and Jan [2] can be summanzed _as follows:

following

the twc-age model of

Partridge

and Barton [1]

they

deal with _a

population consisting

^of individua of _ages 0

l'babies'),

1

l'juveniles'),

2

l'adults').

At each time step

l'generation')

the adults

just

die off and babies

(juveniles)

mature to

juvemles (adults)

with survival

probability

J

(A)

and give birth to

babies,

the

number of babies

being

distributed

exponentially

with _one average

baby

_per parent.

(This

means that there is _no birth in half of the _{cases, one} birth with

probability 1/2,

two births

with

probability 1/4, etc.).

At every

generation

each individuum aise receives mutations somatic mutations which alter the J value of each individuum _to

Jexp(-ne)

_were

considered,

with _n distributed

exponentially

with _average

equal

to u. A

baby

_survives to _a

juvenile

with

probability Jexp(-ne)

but _smce the mutations _are somatic (1.e.

non-heritable)

it

gives

^birth

to babies

having

survival

probability

J and net J

exp(-ne).

The

probability

A with which this

juvenile

_grows to _an adult is determined first from

Partridge-Barton

assumption ^J + A~

= 1

with the unmutated value of J and then it is altered

by

mutations to

Aexp(-pe)

where _p

(*) Permanent address: Department of Physics, University of Calcutta, 92 APC Road, ^Calcutta 700009, India

818 JOURNAL DE PHYSIQUE I N°6

is

distributed, again exponentially,

but _now with average UV. After

growing

_up to _an

adult,

it

again gives

birth to _a

baby having

survival

probability

J. The final result is

that, starting

from _some 10~ _babies

(with

J distributed between 0 and 1 with

equal probability),

after several hundred of

generations,

_one has _an almost time-invariant ratio of

babies, juveniles

and adults.

Also such behaviour is observed for _{a range} of values for _{u, u} and _e, which _are the

only

three parameters of the model. _It does not matter much whether _or not

initially

ail J and A _are

the _same,

This,

Staulfer and Jan

claim, explains

the basic feature of

population

and

ageing

of

living

_orgamsms

subject

to mutations.

Dur first comment _on this work is that

although

it

produces interesting

results from _a

surprisingly simple model,

it incorporates

only

_very weak

hereditary

mutations and in order to be

meaningful,

it must include

hereditary

mutations.

Therefore,

_we first included

hereditary

mutations in their model and found that the

previous

results _are

qualitatively reproduced.

The

hereditary

mutations _were added _over the somatic mutations

by replacing permanently (at

_every

generation)

J values

by

Jn ₌

Jexp(-eh). Thus,

the _new J value is _now

changed,

_as

before,

to Jn

exp(-ne) by

somatic mutations and the babies survive to

juveniles

with

probabil- ity

Jn

exp(-ne)

and then

give

^birth to _new babies

having

_a

juvenile

survival

probability equal

to Jn. The adult survival

probabilities

_are

given

_as before

by

An

exp(-pe)

where An is related to J via the

Partridge-Barton equation.

We have worked with two versions. In version _{a we}

take the

Partridge-Barton equation

_as J ₊

AS

= 1 and in _version b _we take it _as Jn +

AS

₌ 1.

Thus in version b the

hereditary

mutation affects the adult survival

probability

in the _same

generation,

^while in version _{a one more}

generation

_is needed for this elfect _as

An

_sees the elfect of

hereditary

mutations that has

changed

J in the

previous generation.

The parameter _eh was

taken to be

positive l'good' mutations)

and

negative l'bad'mutations)

with

equal probability.

When it is

positive (negative)

its

precise

value _was chosen

randomly

_{as any} number between 0 and _eu

i-fi

Thus _our model _now has rive parameters: _{u, u, e, eu, fi,} and _we have studied

several

values, keeping

_e

= 0.01. We have

always

started with J values that _were distributed

homogeneously

between 0 and 1. Results _{are as} follows:

il)

If _one varies _fi and

keeps

ail other parameters constant, ^{then for low values of} fi the

population

becomes

stationary

alter several hundred

generations,

but for

high

values of _fi it

decays

to _zero. The

boundary

between these two types ^of behaviour is shown in

figure

1 which looks like _a

'phase diagram'. (Versions

_a and b dilfer

only quantitatively.)

(2)

The adult survival

probabilities (Fig. 2b) depend only

_on the

product

_UV and not _on the individual factors _u and _u for fixed ~alues

off,

_{eu, fi,} but this is not the _case for the

juvenile

survival

probabilities (Fig. 2a).

The

phase diagram (Fig. l)

also

depends

_on the mdividual factors _u and _u.

(3)

A set of pararneter ^{values that results in}

higher juvenile

survival

probabilities

than those of another _set has at the _same time adult survival

probabilities

that _are lower.

(4)

As

regards figure

2, the versions _a and b

give qualitatively

similar

results, only

there is the

quantitative

dilference that J for version _a is

larger

than that for version b

(with

ail

parameters

unchanged)

and for A the trend is _reverse.

(5)

For _eh

= 0 the

population

dies _out

(see Lynch

and Gabriel [4]).

Thus, _we find that

introducing hereditary

mutations in the work of Staulfer and Jan

(with

reasonable values of

parameters)

_{one gets}

meanigful

results. In _view of this _we

attempted

next to construct a

simple

model which

reproduces

ail the essential features of the Staulfer and Jan

model, incorporates hereditary

mutations and does _away with the parameter u which _so far has been chosen rather

arbitrarily (see Lynch

and Gabriel [4] for

biological estimates).

Thus,

_{wg now}

ignore

^{somatic mutations and} fluctuations in birth

(precisely

_one ^birth _per

generation)

and _assume that each individuum _{carnes one} J and _one A value. The J

IA)

value is relevant

only

when it grows from _a

baby (juvenile)

to _a

juvenile (an adult).

When it

D.oB

* + typea

o.07

'YP~^b

0.06

or

O.oS

0.04

+

0.

0 5 ₁

u

820 JOURNAL DE PHYSIQUE I N°6

o

O.g

~

~

X x

) ₊

+

+ _x

fi

+

x

Î

0.8

(

~ m

. .

+

~ ~~

i

~'~

ÎÎ=002

1

. E~r0⁰³

(

à à à

. E =0 05 + E~=0.10 0. 6

x EÎr015

.

Jihco<

0.5

0 20 40 60 80 100

uv

a)

O.B . . E~=0.01

à E~r0⁰²

~

. E~r0.03

. " E~=0.05

à ₊ £~=0¹⁰

~ m , x E~=0.15

1

~ ~

~

Aihoe, E

. .

É 8 +

Î

x

')

+^~

)

~

x .

1 0.4 m

+

#

~~

0 20 40 60 80 100

1<v

b)

Fig. 2. Juvenile and adult survival probabilities J and A _as

a function of _UV for _u

= 10, _{e =} 0.01, _ej

=

0.1. We staff with 10~ babies and _{carry over} to 300 generations. This plot is for type _a, but type b

dilfers only qualitatively. For

u = 2 figure ^{2b would be similar but} figure 2a will change

(see text).

Jtl~~~r and Atheor _are the fines predicted by the formulae J

= 0.935

Iii

+ Ve) and A

= 0.505

/(1+

_vue

derived by Staulfer and Jan [2] on the basis of _some simple arguments.

o.08

u=10.Ù

0.06

S 0.02 1

.

* . decaying

0.00 E

&)

o

o 0.9

o

~ .7

oJ .

1

°.~ .

*

O.I

O.1 0.2

~i<

b)

Fig.

ith

be (4

10~,10~,2000). Some data points were checked with (no, nt, T) = ₍₄ x 0~,10~,3000). The

number nt is the _which ixes

the attern of'food restriction' (see text).

822 JOURNAL DE PHYSIQUE I N°6

decays

with

generations

and for the other values the

population

increases

monotomcally (with-

out food

restriction),

_or attains _a

stationary

value

(with

food

restriction).

The

stationary

value for survival

probabilities

_are in tum

independent

of the

input

pararneters.

This,

_we

believe, might

be _an indication in support ^of some agemg theories based _{on mutation} accumulation _[GI.

Acknowledgements.

We _are indebted _to Dietrich Staulfer for

introducing

_us to the

subject

of

ageing

and for his active _interest in _our work and to Naeem Jan for

suggesting

the search for fixed points.

S-V- thanks the

Rechenzentrum, Cologne University,

for _{a generous} grant of computer ^time and

general

support. S-D- thanks SFB 341 for support and HLRZ St.

Augustin

for _comuter

time.

References

[1] Partridge L. and Barton N-H-, Nature 362

(1993)

305.

[2] Staulfer D. and Jan N., preprint

(1993);

see also Jan N., prepnnt

(1994).

[3] Ray T.S., J, Stat. Phys. 74

(1994)

929.

[4] Lynch M. and Gabriel W., Evolution 44

(1990)

1725.

[Si Murray J. D., Mathematical Biology

(Spnnger-Verlag,

Berlin, Heidelberg, 1989) _p. 2.

[6] ^{Rose M.R. and} Charlesworth B., Nature 287

(1980)

141.