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Live 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. [1] U. Herbig, W. A. Jobling, B. P. Chen, D. J. Chen, and [9] J. B. Coe and Y. Mao, Phys. Rev. E 69, 041907 (Apr J. M. Sedivy, Molecular Cell 14, 501 (May 2004) 2004), http://link.aps.org/doi/10.1103/PhysRevE. 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Wong, BF02180147 G. Sigurdsson, G. B. Walters, S. Steinberg, H. Helga- [7] D.Stauffer,BioinformaticsandBiologyInsights1(2007) son, G. Thorleifsson, D. F. Gudbjartsson, A. Helgason, [8] J. B. Coe and Y. Mao, Phys. Rev. E 67, 061909 (Jun O. T. Magnusson, U. Thorsteinsdottir, and K. Stefans- 2003), http://link.aps.org/doi/10.1103/PhysRevE. son,Nature488(Aug2012),doi:“bibinfodoi10.1038/na- 67.061909 ture11396

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