Table Of ContentLive Young, Die Later: Senescence in the Penna Model of Aging
Avikar Periwal
Montgomery Blair High School, 51 University Blvd East, Silver Spring Maryland 20901∗
(Dated: December 11, 2013)
Cellularsenescenceisthoughttoplayamajorroleinage-relateddiseases,whichcausenearly67%
of all human deaths worldwide. Recent research in mice showed that exercising mice had higher
levels of telomerase, an enzyme that helps maintain telomere length, than non-exercising mice. A
commonlyusedmodelforbiologicalagingwasproposedbyPenna. Iproposetwomodificationsofthe
Penna model that incorporate senescence and find analytical steady state solutions following Coe,
Mao and Cates. I find that models corresponding to delayed senescence have younger populations
that live longer.
3 I. INTRODUCTION
1
0 From early alchemists looking for the elixir of life, to
2
modern day researchers, humans have always wanted to
n understand aging. As people age, their cells go through
a replication cycles. Each replication reduces the length
J
of the telomeres in the cells. If a cell’s telomeres are
8 too short, it may not be able to replicate[1]. Cells that
1
can no longer replicate are termed senescent. Cellular
senescence is thought to play a major role in age-related
]
E diseases, which cause nearly 67% of all human deaths
P worldwide[2]. Recent research in mice showed that ex-
. ercising mice had higher levels of telomerase, an enzyme
o
thathelpsmaintaintelomerelength,thannon-exercising
i
b mice[3]. In humans, runners had longer telomeres than FIG. 1. The possible paths until death, after reaching l, in
-
q non-runners[3]. Since shortened telomeres are thought the SP.
[ to be related to death, this research would seems to in-
dicatethatpeoplewhoexerciselivelongerlives. Popula-
2
tion studies do indeed show this, however, most studies the simulation. A 1 in the string represents a mutation,
v
show that only the mean lifespan increases in exercising and a 0 means no mutation. If an individual has gone
3
9 populations, not maximum lifespan[4, 5]. through T 1s, then it dies. Each individual can have
0 A commonly used model for biological aging was pro- offspring, with probability b. The child’s bit-string is
1 posed by Penna in 1995[6, 7]. The model looks at death derived from the the bit-string of the parent, where each
1. fromamutationaccumulationstandpoint. Inthelast10 0 has probability m of becoming a 1. The Penna model
0 years,papershavebeenpublisheddescribingmethodsfor ignores positive mutations, because they are rare. The
3 findingtheagedistributionscreatedbythePennamodel lengthofthebit-stringprovidesahardlimitforlifespan.
1 without actually simulating the model[8–10]. However, In this paper, I will work with only the T =1 case.
v: the research on senescence indicates that age-related In the first proposed modification, which we will call
i deathisnotcausedbymutationaccumulation,butrather thesenescentPenna(SP)model,eachindividualcanonly
X
by an inability to reproduce after a number of cycles. I get one disease, which is essentially the beginning of ag-
r propose two modifications of the Penna model that use ing. After the individual starts to age, it has probability
a
thismechanisminsteadofamutationaccumulation, and p of staying alive to reproduce at each time step. The
show that a mean lifespan can increase without affecting maximum number of replication cycles is M. This is
the maximum lifespan, which cannot be done by chang- the Penna model, except instead of definitely dying, the
ing parameters in the original Penna model. individual has only a probability of dying. People who
exercise generally have longer telomeres, so once senes-
cence starts, they have a smaller probability of dying.
II. CALCULATIONS Higher values of p represent exercising populations.
The first 1 in an individual’s bit string is the age, l, at
ThePennamodelassignsabit-stringtoeachindividual which senescence begins. The number of people alive at
in the population. Each bit corresponds to a timestep of timej,withagex,isnj(x,l,m)=pnj−1(x−1,l,m−1)if
x>l,wherepistheprobabilityoflivingaftersenescence,
and m is the time since the inception of senescence. If
x≤l,thenn (x,l)=n (x−1,l). Ifbistheprobability
j j−1
∗ [email protected] ofbirth,ande−β istheprobabilityofanindividualgoing
2
unmutated,thenthenumberofchildrenborninthenext dies,butheretheindividualhasonlya(1−p)probability
time step with disease acquisition time l, n (0,l,0) is of dying. The number of individuals who die at age t is
j+1
given by just
(cid:88)(cid:88)
n (0,l,0)=be−βl n (x,l,m) t
j+1 j (cid:88)
D(t)=pMn(t−M)+ (1−p)pt−xn(x), (10)
x m
(cid:88)(cid:88)(cid:88)
+(1−e−β)be−βl n (x,l(cid:48),m) x=t−M+1
j
l(cid:48)>l x m
assuming t ≥ M. The first term ensures that when t =
(1)
l+M, all the remaining people alive with n(l) die, not
Since m is just the maximum of 0 and x−l,
just a proportion of 1−p.
l−1 M+l−1 ItisalsopossibletocombinetheoriginalPennamodel
(cid:88)(cid:88) (cid:88) (cid:88)
nj(x,l,m)= n(x,l)+ n(x,l) (2) with the modification described previously, making a
x m x=0 x=l combination Penna (CP) model. Some people have a
genetic tendency for heart disease, or stroke[11]. These
where M is the maximum number of replication cy-
diseasesarenotdirectlyaffectedbysenescence. Inacom-
cles allowed after disease. Assuming a steady state,
bination of the two models, there would be two classes
n (x,l)=n (x,l). Therefore,
j+1 j of death, one instantaneous, as in the original model,
and one probability based, as in the proposed modifica-
l−1
(cid:88) tion. Then each individual is defined in terms of l and
n(x,l,0)=l·n(0,l,0), (3)
l , where l is the time of a probabilistic mutation and
x=0 P
l is the time to a Penna mutation, or instant death.
P
since each x has the same number of people. Since n (0,l,l ) can come from n (x,l(cid:48),l(cid:48) ) where l(cid:48) ≥l and
j+1 P j P
n (x,l)=pn (x−1,l)forx>l,andsincen (x,l)= l(cid:48) ≥ l . Note that l cannot be greater than l . There-
j j−1 j+1 P P P
n (x,l) in a steady state fore,
j
M(cid:88)+l−1n(x,l)=n(0,l)(cid:88)M px =p1−pMn(0,l) (4) nj+1(0,l,lP)=be−αle−βlP[(cid:88)nj(x,l,lP)
1−p x
x=l x=1 (cid:88) (cid:88) (cid:88)
+(1−e−α)(1−e−β) n (x,l(cid:48),l(cid:48) )
j P
Defining q as
l l(cid:48)>llP(cid:48) >lP x
(cid:88)(cid:88)
1−pM +(1−e−β) n (x,l(cid:48),l )
q =l+p , (5) j P
l 1−p l(cid:48)>l x
(cid:88) (cid:88)
+(1−e−α) n (x,l,l(cid:48) )],
and n(l)=n(0,l), eq. 1 can be simplified to j P
lP(cid:48) >lP x
n(l) (cid:88) (11)
0=be−βln(l)− +(1−e−β)be−βl n(l(cid:48)) (6) where e−α and e−β are the probabilities of not getting
q
l
l(cid:48)>l+1 a senescence or death mutation, respectively. The death
distribution is given by probabilistic and instantaneous
Writing the same equation for l+1, some algebra leads
deaths,
to
be−βl−1/q t
n(l+1)=n(l) l (7) D(t)= (cid:88) pt−xn(x,t)
be−β(l+1)−eβ/q
l+1
x=t−M
This equation leads to some limiting cases, in order to t (12)
(cid:88) (cid:88) (cid:88)
maintain a steady state. Neither the numerator, nor the + pt−x(1−p)n(x,l )
P
denominator should vanish in eq. 7. lP>tl<tx=t−M+1
+pMn(t−M,l )
1 P
q < , (8)
max 1−e−β
As time passes in the Penna model, the mutations
(and l =q −p1−pM) and slowly move downwards. However, there is no evolution-
max max 1−p
ary pressure preventing an individual with l = 0 from
1 reproducing. Since there are no positive mutations, my
b= (9)
q e−βl ansatz is that in a steady state, l=0 for all individuals.
max
If l = 0, the individual does not reproduce, so there
P
However,eq. 7onlygivesthetimeofsenescence,notthe is an evolutionary pressure for l > 0. If this happens,
P
lifespan. IntheoriginalPennamodel, atl theindividual thenanytermssummingoverdifferentvaluesoflvanish,
3
FIG.2. Highervaluesofpresultinearliertimesofsenescence FIG.3. ThedeathcurveintheSPisashiftofthesenescence
in the SP. In this picture, M =5. curve,withtheamountshiftedvaryingonp. HereM =5and
p=.8.
so
(cid:88) (cid:88)
n (0,l,l )=be−αle−βlP[ n (x,l,l )
j+1 P j P
lP(cid:48) =lP x (13)
(cid:88) (cid:88)
+(1−e−α) n (x,l,l(cid:48) )].
j P
lP(cid:48) >lP x
Simplifyinginthesamemannerasinthepreviousmodel,
the steady state form is the same as eq. 7, except
1−plP
q =p . (14)
l 1−p
However, since l=0, the time until senescence is always
0,eq. 12isnotneeded. Therecursiveformjustgivesthe
age of death.
III. RESULTS
FIG. 4. Higher values of p have a higher proportion of their
Interestingly, in the SP model higher values of p show population reach larger ages, but that proportion difference
a “younger population” (fig. 2). With lower p values, an is countered in the middle of the death distribution. M =5
individualwithlowlwillnothaveasmanyopportunities
to reproduce, since chances are it will die out soon. This
provides an evolutionary pressure for higher values of l. whenanylivingmembersofthepopulationareundergo-
However,ifpishigh,thenindividualswithlowl canstill ing senescence. Figures 4 and 5 show the differences in
reproduce. percentages, which have to add up to 0. The increased
While senescence begins later for populations with proportion of older individuals for higher values of p has
lower p, death comes earlier once senescence is reached. to be balanced by a reduced proportion of the popula-
The average death time for n(l) is l+(1−p)+2p(1− tion at lower ages. The jump in the differences of death
p)+3p2(1−p)+...+(M +1)pM, where the last term proportions early in figs. 4 and 5 is caused by M. For
has no 1−p factor, since everybody left alive has to die. largervaluesofp,the1−ptermineq. 10issmallenough
Higher values of p result in later deaths, as would be ex- to make the pt−x term negligible. However, once M is
pected. The later deaths are more apparent after l , reached, the final term has no (1−p) factor, causing the
max
4
FIG. 5. This shows the same break as fig. 4, but at age 10, FIG. 7. The death distributions for different values of p
since M =10 in the SP here. look approximately the same, but there is a slightly higher
proportion of people alive at later times for higher values of
p.
FIG. 6. The differences in the death distribution curves of
the SP of M =5 and M =10, for different values of p.
FIG.8. ThisisthedeathdistributionintheCP,fordifferent
values of p. Notice that even a slight change in p has a large
jump. change on the death distribution.
The SP model does not show an increase in the mean
forhighervaluesofp,butitdoesshowthattheprobabil-
ity of living to a higher age is greater. Even though the portionofpeoplealiveatahigherageisgreaterforhigher
probability of living to a high age is greater, the maxi- valuesofM. Thebreaksinfig. 5arecausedbythesame
mumageforbothpopulationswithhigherandlowerpis mechanism as the breaks in figs. 3 and 4, except instead
stilll +M. HighervaluesofM pushthetimeofsenes- ofthedeathscomingatonespecificM,theycomeatthe
max
cence further forward, since there is less evolutionary in- two values of M, 5 and 10.
centive for an individual to have a higher l. However, TheCPmodelshowsadifferentstructurethantheSP
just like p, higher values of M also afford a longer time model. It shows a quicker increase in age expectancy,
until death, balancing out the earlier senescence times. and a slower decrease. The combination requires a high
Since higher values of M allow for a longer life, the pro- p, because otherwise q degenerates rapidly to p . If p
1−p
5
is too low, then the CP model will not be stable without tions. The maximum age of two populations with differ-
a Verhulst factor, ent p and the same M is the same, but the exercising
population has a higher chance of reaching later ages.
N(t) What is interesting, is that this model shows that the
V =1− , (15)
N exercising population will also be younger. The times
max
of senescence for high p are lower than low p, yet the
whereN(t)isthenumberofpeoplealiveattimet,N
max probability of reaching a high age is greater.
is the maximum allowed number of people, and V is the
probability that an individual survives a timestep. This
also explains why even small differences in the value of p
show large differences in the age distribution curve.
The Penna model is a tool used to help us under-
stand population dynamics. My modifications of the
IV. CONCLUSION Penna model take into account senescence of a popula-
tion,whichisacriticalpartoftheagingprocess,andhelp
Inthispaper,Ishowedthatsimplemodificationstothe to explain the changes in lifespan observed in exercising
Penna model allow for shifts in the lifespan distribution andnon-exercisingpopulations. Byadjustinge−β,p,and
without changing the maximum lifespan. Higher values M, these should be able to fit actual data. Further im-
ofpintheSPresultinyoungerpopulations,buttheydie provements to the models could take into account recent
later. research in autism which suggests that e−β is actually a
Thinking about the original context for this modifica- function of time[12], and also looking at positive muta-
tion, higher values of p can represent exercising popula- tions.
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