Aging: Modeling Time

1. Aging: Modeling Time This thing all things devours:Birds, beasts, trees, flowers;Gnaws iron, bites steel;Slays king, ruins town,And beats high mountain down Tom Emmons

2. Outline • Start Simple – only death • Add properties • Birth • Age • Life Stages • Some real life examples

3. Laws 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 age

4. Discreet 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 σj

5. The 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)

6. 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 ratio

7. Continuous Models • Two directions to go: • Stages aren’t continuous • Reproduction of an animal population • Transitions don’t happen at discreet intervals • Differentiation of cells

8. A 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 play

9. A simple model: the graphs Graphs from L. Edelstein-Keshet Et Al.(2001)

10. An example: Stem Cells

11. Telomeres • 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 loss

12. A model • Add reproduction to our previous continous model • “Death” is differentiation

13. Setting 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)

14. 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!

15. 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 behavior

16. References • 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.

17. Images • http://www.exploredesign.ca/blog/wp-content/uploads/2007/09/gollum.jpg • www.srhc.com/babypics/Baby/pages/Images/baby.jpg • http://www.robertokaplan.at/images/old-woman-madeira.jpg