Aging: Modeling TimeOutlineLaws of MortalityDiscreet ModelsThe math (I didn’t think pictures would substitute)This is trivial… why do we care?Continuous ModelsA simple ModelA simple model: the graphsAn example: Stem CellsTelomeresA modelSetting up the mathPredictionsConclusionsReferencesImagesAging: Modeling TimeTom EmmonsThis thing all things devours:Birds, beasts, trees, flowers;Gnaws iron, bites steel;Slays king, ruins town,And beats high mountain downOutline•Start Simple – only death•Add properties–Birth–Age–Life Stages•Some real life examplesLaws of Mortality•The Gompertz equation (1825)–t is the time–N(t) is population size of a cohort at time t–γ(t) is the mortality–A is the time rate of increase of mortality with age0.5 1 1.5 20.20.40.60.81Discreet Models•Instead of death, cells die or move to discreet next phase. Each phase has unique birth rate•Assumptions:–L is maximal lifespan–n is number of distinct classes–P0(t), P1(t),…, Pn(t) denote the number of females in a population age class–Birth only in age class 0–Age dependent mortality μj–Age dependent birth rate σjThe math (I didn’t think pictures would substitute)•Time t measured in units L/n•Predictions:–Without birth and death, cohort ages with time–Exponential growth without death and constant birth–Expential decay with constant death rate–If both mortality and birth are constant, the population scales by factor of P(α+1-μ)This is trivial… why do we care?•Our model can be handled with Linear Algebra!!!•Letting M be a matrix of coefficients, we can write:•The growth rate becomes the dominant eigenvalue•The population approaches a well-defined ratioContinuous Models•Two directions to go:–Stages aren’t continuous•Reproduction of an animal population–Transitions don’t happen at discreet intervals•Differentiation of cellsA simple Model•Start Simple: No birth, No death–Total number of cells is constant–Letting D be the mean differentiation stage, –Each division class has a time of maximum population–The age distribution at any time has a peak, but the distribution widens with time–These results assume a final stage doesn’t come into playA simple model: the graphsGraphs from L. Edelstein-Keshet Et Al.(2001)An example: Stem CellsTelomeres•Ends of chromosomes, containing repeats of (TTAGGG)•Cell division results in decreased length–Humans lose 50-200 (average 100) bp•Some cells (germline and some somatic cells) have telomerase or other mechanisms to avoid this lossA model•Add reproduction to our previous continous model•“Death” is differentiationSetting up the math•Let Sn be the number of stem cells that have undergone n divisions•Let p be the rate of self renewal and f the rate of differentiation–One cell comes from an f event (differentiation)–Two cells come from a p event (self renewal)Predictions•Total number of cells increases with growth rate p•Mean telomere length decreases by roughly•If we know the growth rate and mean change in length, we can find p and f!Conclusions•By slowly building models of aging up, we can make real predictions about our systems and also backtrack information out•Models must ultimately move to non-linear regimes to better describe actual behaviorReferences•Edelstein-Keshet, Leah, Aliza Esrael and Peter Lansdorp. “Modelling Perspectives on Aging: Can Mathematics Help us Stay Young?” 2001 Academic Press•Caswell, H. (2001). Matrix Population Models: Construction, Analysis, and Interpretation, 2nd edn. Sunderland, MA: Sinauer Associates.Images•http://www.exploredesign.ca/blog/wp-content/uploads/2007/09/gollum.jpg•www.srhc.com/babypics/Baby/pages/Images/baby.jpg
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