Transcript Aging: Modelling Time

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
Outline
• Start Simple – only death
• Add properties
– Birth
– Age
– Life Stages
• Some real life examples
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
1
0.8
0.6
0.4
0.2
0.5
1
1.5
2
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
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)
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
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
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
A simple model: the graphs
Graphs from L. Edelstein-Keshet Et Al.(2001)
An example: Stem Cells
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
A model
• Add reproduction to our previous
continous model
• “Death” is differentiation
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)
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 behavior
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.
Images
• http://www.exploredesign.ca/blog/wpcontent/uploads/2007/09/gollum.jpg
• www.srhc.com/babypics/Baby/pages/Imag
es/baby.jpg
• http://www.robertokaplan.at/images/oldwoman-madeira.jpg