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>
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Phys.
J.
I
France
(1994)
4
817-822
3UNE
1994,
817
PAGE
Classification
Physics
Abstracts
05.40
87.10
Communication
Short
A
ageing
for
mortel
Stefan
Vollmar
of
Institute
(Received
We
show
We
mutations
aise
Physics, Cologne University,
April 1994, accepted
13
inherited
investigate
hereditary
mutations
Subinay Dasgupta (~,*)
and
Theoretical
Abstract.
nous
with
trie
19
50937
KüIn,
Germany
Apnl 1994)
simple model that incorporates a balance of helpful and deletea
by future
leads to an
'explanation' of biological ageing.
generations,
stability hmit beyond which the populations decay to zero.
that
ages, biologists and ecologists have been trying to explain why living beings age and why
populations of some
show interesting
temporal
behaviour
je-g-, remain constant)
species
found
this
helpful
Recently,
has
been
that
in
aise
it
connection
it
is
to take
[1].
very
recourse
of
computational
physics,
mainly
those
suited
for
the
techniques
to
systems made of a huge
Staulfer
number
of interacting
elements.
In one of such works,
and Jan [2] have
shown
that
'discrete'
model
reproduce
elfects.
simple
incorporating
somatic
mutations
ageing
can
some
a
Ray [3] continued this approach by investigating the dynamics.
The present
communication
aims at
certain
investigations related to this work. We present
of
results
which
be
the
subject
future
research.
In a nutshell, the work by Staulfer
and
can
some
summanzed
follows:
following
the
model
of
Partridge
and
Jan [2] can be
Barton
twc-age
as
of ages 0 l'babies'), 1 l'juveniles'),
individua
[1] they deal with a population consisting of
l'generation') the adults just die off and babies (juveniles)
At each
time step
2 l'adults').
(adults) with survival probability J (A) and give birth to babies, the
mature
to juvemles
(This
number
of babies being
distributed
exponentially with one average baby per parent.
1/2,
half
of
that
there
birth
the
birth
with
probability
births
is
in
two
no
cases,
one
means
probability 1/4, etc.). At every generation each
individuum
aise
with
receives
mutations
Jexp(-ne) were considered,
which alter the J value of each
individuum
somatic
mutations
to
exponentially with average equal to u. A baby survives
with n
distributed
with
to a juvenile
Jexp(-ne) but smce the mutations
probability
somatic (1.e. non-heritable) it gives birth
are
probability J and net J exp(-ne). The probability A with which this
babies having survival
to
first from
Partridge-Barton
juvenile grows to an adult is determined
assumption J + A~
1
Aexp(-pe) where p
by mutations
with
the
unmutated
value of J and
then it is
altered
to
For
the
=
(*)
700009,
Permanent
India
address:
Department
of
Physics,
University
of
Calcutta,
92
APC
Road,
Calcutta
JOURNAL
818
distributed, again
again gives birth
is
it
to
exponentially, but now
survival
a baby having
with
I
average
N°6
J.
growing
After
UV.
probability
final
The
result
up
to
adult,
starting
an
that,
is
equal probability), after several
of babies, juveniles and adults.
one
an
Also such
behaviour is observed for a range of values for u, u and e, which are the only three
much
whether
initially ail J and A are
parameters of the model. It does not
matter
not
or
Staulfer
the
This,
and
Jan
claim,
explains
the
basic
feature
of
population and ageing of
same,
living orgamsms subject to mutations.
Dur first
this
work is that although it produces
interesting results from a
comment
on
surprisingly simple model, it incorporates only very weak hereditary
and in order to
mutations
include
hereditary
Therefore, we first included hereditary
be meaningful, it must
mutations.
model and found that the previous results are qualitatively reproduced.
mutations
in their
added over the somatic
mutations
mutations by replacing permanently
The hereditary
were
(at every generation) J values by Jn
Jexp(-eh). Thus, the new J value is now changed, as
mutations
and the babies
before, to Jn exp(-ne) by somatic
survive to juveniles with probability Jn exp(-ne) and then give birth to new babies having a juvenile survival probability equal
survival probabilities are given as before by An exp(-pe) where An is related
to Jn. The adult
Partridge-Barton
equation. We have worked with two versions. In version a we
to J via the
Partridge-Barton
equation as J + AS
b we take it as Jn + AS
take the
1 and in
1.
version
affects the adult
probability in the same
mutation
survival
Thus in version b the hereditary
generation, while in version a one more generation is needed for this elfect as An sees the elfect
that has changed J in the previous generation.
The
of hereditary
mutations
parameter eh was
l'bad'mutations) with equal probability.
taken to be positive l'good' mutations) and negative
chosen
randomly as any number
When it is positive (negative) its precise value
between
was
from
10~ babies (with J
PHYSIQUE
DE
some
hundred
generations,
of
between
distributed
almost
has
0 and 1 with
time-invariant
ratio
=
=
0 and
i-fi
eu
Thus
homogeneously
il)
our
values, keeping
several
If
population
decays to
between
varies
one
e
fi
becomes
model
now
has rive
0.01.
We
have
=
0 and
parameters:
always
Results
1.
keeps ail
stationary alter
and
=
parameters
hundred
several
u,
u, e,
with
J
eu,
and
fi,
values
that
have
we
studied
distributed
were
follows:
as
are
other
started
then
constant,
generations,
for
but
for
low
values
high
values
of fi
of
the
fi
it
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
probabilities (Fig. 2a). The phase diagram (Fig. l) also depends on the mdividual
survival
factors
(3)
of
and
u
pararneter
has
set
quantitative
unchanged)
parameters
For
Thus,
eh
we
reasonable
=
the
dilference
versions
that
find
values
of
parameters)
a
J for
for A the
and
population
that
introducing
0 the
in
survival
adult
time
same
higher juvenile survival probabilities than those
probabilities that are lower.
similar
and b give qualitatively
results, only there
than
that for version b (with ail
version
a is larger
results
that
values
the
at
regards figure 2,
As
the
(5)
u.
A set of
another
(4)
is
The
zero.
trend
is
reverse.
(see Lynch and Gabriel [4]).
hereditary
in the work of
mutations
gets meanigful results. In view of
one
dies
out
Staulfer
this
we
and
Jan
(with
attempted
next
features of the Staulfer
and Jan
simple model which reproduces ail the essential
the
and
does
with
which
far
mutations
model, incorporates hereditary
parameter u
away
so
estimates).
and
Gabriel
for
biological
arbitrarily
(see
Lynch
rather
has been chosen
[4]
fluctuations
and
birth per
mutations
in birth (precisely one
Thus, wg now ignore somatic
to
construct
generation)
value
is
a
and
relevant
assume
only
that
when
it
each
individuum
grows
from
a
carnes
one
baby (juvenile)
J
to
and
a
one
A
value.
The
juvenile (an adult).
J
When
IA)
it
A
N°6
MODEL
FOR
AGEING
IIITH
HEREDITARY
MUTATIONS
D.oB
+
*
type a
'YP~ b
o.07
0.06
O.oS
or
0.04
+
0.
0
u
5
1
819
JOURNAL
820
PHYSIQUE
DE
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N°6
o
O.g
~
)
X
+
x
+
fi
Î
(
x
+
0.8
x
~
m
.
.
+
i
~
+
~'~
~ ~~
1
ÎÎ=0
(
à
à
à
.
0. 6
.
02
E~r0 03
=0 05
.
E
+
E~=0.10
x
EÎr0
15
Jihco<
0.5
20
0
40
60
80
100
uv
a)
O.B
.
.
E~=0.01
à
E~r0 02
E~r0.03
~
.
.
"
à
~
1
E
+
,
m
x
~
.
.
+
8
Î
x
+
')
~
)
1
Aihoe,
~ ~
É
E~=0.05
£~=0 10
E~=0.15
~
.
x
m
0.4
+
#
~~
20
0
40
60
80
100
1<v
b)
Fig.
2.
0.1.
We
dilfers
Juvenile
staff
only
with
and
adult
10~
babies
qualitatively.
Jtl~~~r and Atheor
derived
by Staulfer
are
the
and
For
fines
Jan
probabilities
survival
[2]
and
u
carry
2 figure
predicted by
=
on
the
to
over
basis
2b
the
of
J
300
would
and A
be
formulae
some
as
function
a
generations.
similar
J
simple
=
of
but
0.935
UV
for
plot is
figure 2a
This
Iii
arguments.
+
u
for
will
Ve) and A
10,
=
type
e
a,
=
0.01,
but
ej
=
type b
change (see text).
=
0.505
/(1+
vue
MODEL
A
N°6
FOR
AGEING
WITH
HEREDITARY
MUTATIONS
821
o.08
u=10.Ù
0.06
S
1
0.02
.
.
*
decaying
0.00
E
&)
o
o
0.9
o
~
.7
.
oJ
1
°.~
.
*
O.1
0.2
O.I
~i<
b)
Fig.
ith
be
number
(4
nt
10~,10~,2000).
is
Some
the
data
points
ixes
which
the
attern
were
of'food
restriction'
checked
(see
with (no,
text).
nt,
T)
=
(4
x
0~,10~,3000).
The
JOURNAL
822
decays with
food
for
survival
might
generations
restriction),
out
be
attains
or
indication
I
N°6
other values the population
increases
monotomcally (withstationary value (with food restriction). The stationary value
independent of the input pararneters.
This, we believe,
tum
and for the
probabilities
an
PHYSIQUE
DE
in
a
in
are
of
support
based
theories
agemg
some
on
mutation
accumulation
[GI.
Acknowledgements.
We
indebted
are
active
S-V-
thanks
and
general
Dietrich
to
in
interest
work
our
to
Rechenzentrum,
the
S-D-
support.
introducing us to
suggesting
Cologne University, for a
Staulfer
and
thanks
for
Jan for
Naeem
SFB
for
341
and
support
the
the
subject of ageing
and
for
points.
search
fixed
References
Partridge
[2]
Staulfer
see
also
L.
and
D.
and
Jan
N.,
[3] Ray T.S., J,
[4] Lynch
M.
[Si Murray
[6] Rose
J.
M.R.
Barton
Jan
N.,
prepnnt
N-H-,
preprint
Nature
362
(1993)
305.
(1993);
(1994).
Phys. 74 (1994) 929.
Gabriel W.,
Evolution
44 (1990) 1725.
Mathematical
Biology (Spnnger-Verlag,
Stat.
and
D.,
and
Charlesworth
B.,
Nature
287
(1980)
Berlin, Heidelberg, 1989)
141.
his
time
grant of computer
generous
HLRZ St. Augustin for
comuter
time.
[1]
for
p.
2.