Diversity Within Species

1
Diversity Within Species

• Erik Rauch Yaneer Bar-Yam
• New England
• Complex Systems Institute

2
Measuring diversity

• Diversity is often measured by counting species.
• Destruction of habitat area causes loss of
  species.
• Species area relationship
• S Az
• z ? 0.25

3
Genetic diversity in a species

• Number of genetic differences between pairs of
  organisms.
• Total number of genetic differences of the whole
  population.
• We study the structure of genealogical trees to
  investigate genetic differences
• ---coalescent theory.

4
Methodology

• Simulations and Coalescent theory approach
• Coalescent theory references
• R. R. Hudson, Oxford Surv. Evol. Biol. 7, 1
  (1990).
• N. H. Barton, I. Wilson, Phil. Trans. Roy. Soc.
  B 349, 49 (1995).
• B. K. Epperson, Evol. Biol. 27, 95 (1993).
• M. Notohara, J. Appl. Prob. 38, 1 (2001).
• J. Wakeley, Genetics 161, 873 (2002).

5
Genetic diversity has spatial structure in many
populations

• J. C. Avise, Phylogeography The History and
  Formation of Species (Harvard University Press,
  Cambridge, MA, 2000).T. Zerjal et al., Mol.
  Biol. Evol. 18, 1077 (2001).M. Richards, V.
  Macaulay, A. Torroni, H. J. Bandelt, Am. Jour.
  Hum. Genet. 66, 262 (2002).M. D. Bowen et al.,
  J. Virol. 74, 6992 (2000).J. C. Cho, J. M.
  Tiedje, Appl. Env. Microbiol. 66, 5448 (2000).K.
  Schiemann, T. Tyler, B. Widen, Plant Syst. Evol.
  225, 119 (2000).S. W. Lee, W.Y. Choi, W. W. Kim,
  Z. S. Kim, Silvae Genet. 49, 124 (2000).C. L.
  Elderkin, P. L. Klerks, J. N. Amer. Benthological
  Soc. 20, 595 (2001).G. Rowe, T. J. C. Beebee, T.
  Burke, Oikos 88, 641 (2000).B. Goossens, L.
  Chikhi, P. Taberlet, L.P. Waits, D. Allaine, Mol.
  Ecol. 10, 41 (2001).A. M. Baker, P. B. Mather,
  J. M. Hughes, Biol. J. Linnaean Soc. 72, 333
  (2001).G. Clausing, K. Vickers, J. W. Kadereit,
  Mol. Ecol. 9, 1823 (2000).

6
Genealogical Tree
7
2-dimensional Relatedness
8
Pairwise relatedness
well mixed
9
Uniqueness of an individual

• Distribution is a power law.
• Much of the diversity is located in only a few
  individuals.

10
Diversity Fluctuations
11
Losses of diversity.

• Distribution is a power law.
• Large fluctuations up to a fraction (1/3) of the
  total diversity.

12
Structure of Genealogical Tree

• Consider genealogical tree as a backwards
  process. Each organisms parent is in a random
  direction and the ancestry is a random walk. Two
  organisms have the same parent when their random
  walks step to the same location.
• The population is a set of random walks starting
  from full set of points in space. Gradually
  coalesce until only one is left.

13
Number of walkers

• (Use result for diffusion process with absorption
  at the origin)
• M. Bramson, D. Griffeath, Z. fuer Wahr. 53, 183
  (1980).
• Well-mixed L(t0-t)1/(t0-t)
• In 2-d L(t0-t)log(t0-t)/(t0-t)
• In 1-d L(t0-t)1/(t0-t)1/2

14
Global Pseudomonas Samples Dendogram

• J. C. Cho, J. M. Tiedje,
• Appl. Env. Microbiol.
• 66, 5448 (2000).

15
Locations of samples of Cho Tiedje data
16
Comparison of Experiment with Theory
L(t0-t)
t0-t
17
Short time approximate scaling (spatial
undersampling)

• L(t0-t) L0 - c log(t0-t)

L(t0-t)
t0-t
18
Total Genetic Diversity

• Total branch length
• B ?D(t0-t)
• (upper limit is time to single ancestor, T).
• If the mutation rate is low this is proportional
  to the Genetic diversity.
• D(B) µB
• Otherwise (more exact)
• D(B) (µ/µ0)(1-eµ0B)

19
Time to common ancestor

• Two dimensions
• L(t0-t)Alog(t0-t)/(t0-t)
• L(T)1
• ?
• TAlog(A)

20
Genetic Diversity-Area Relationship

• The dependence of diversity on area is
• B(A) A(log(A))2

21
Biodiversity increase with area
Genetic Diversity area relationship
Species Diversity area relationship
22
One-dimension

• High diversity
• L A/(t0-t)1/2
• ?
• TA2
• ?
• B(A)A2
• Fractal dimension dlt2
• B(A) A2/d

23
Conclusions

• Spatial structure without migration events or
  imposed boundaries.
• Scale free distribution of uniqueness.
• Scale free distribution of fluctuations.
• Scale free distribution of pairwise distance.
• Genealogical tree scaling and sampling behavior
  matches experiment.
• Genetic diversity area relationship gt
• Loss of area gt dramatic loss of diversity.