Holism

2. Holism • “The whole is greater than the sum of its parts” - Aristotle

3. Holism and Atomism • “The whole is greater than the sum of its parts” - Aristotle • “The whole is less than the sum of its parts” - Edward Lewis

4. Two triangulations of the bipyramid • “The whole is greater than the sum of its parts” - Aristotle • “The whole is less than the sum of its parts” - Edward Lewis

5. AB ? aB AB ? fitness AB ? ab Ab genotype Epistasis • Two-locus two-alleles: ab aB Ab ABwith fitness landscape wab waB wAb wAB

6. positiveepistasis wab+wAB > wAb+waB negativeepistasis wab+wAB < wAb+waB Epistasis • Two-locus two-alleles: ab aB Ab ABwith fitness landscape wab waB wAb wAB aB wab+wAB = wAb+waB AB fitness ab Ab genotype

7. u < 0 u = 0 u > 0 Geometric perspective • Two-locus two-alleles: 00 01 10 11with fitness landscape w00 w01 w10 w11epistasisu = w00 + w11 – w01– w10 Two generic shapes of fitness landscapes

8. Populations and the genotope • n loci, allele alphabet (or , or …) • Genotype space: • The genotope is the space of all possible allele frequencies arising from . It is the convex polytope population simplex marginalization map allele frequency space

9. 01 11 00 10 Example: 01 10 11 00

10. Fitness landscapes and interactions • A fitness landscape is a function . • Linear functions have no interactions, so consider theinteraction space • For example: • The interaction space is spanned redundantly by the circuits, i.e., the linear forms with minimal support in . • Hypercubes have natural interaction coordinates given by the discrete Fourier transform.

11. Example 1: 111 001 100 010 One circuit: 000

12. Example 2: Four circuits:

13. Example 3: The vertebrate genotopes Margulies et al., 2006.

14. Example 3: Towards the human genotope HapMap consortium, 2005

15. The shape of a fitness landscape • Extend to the genotope: For all , • The continuous landscape is convex and piecewise linear. • The domains of linearity are the cells in a regular polyhedral subdivision of the genotope. • This subdivision is the shape of the fitness landscape, . populationfitness

16. u < 0 u = 0 u > 0 Fittest populations with fixed allele frequency {00, 01, 10}{01, 10, 11} {00, 01, 10, 11} {00, 01, 11}{00, 10, 11}

17. Two triangulations of the triangularbipyramid • “The whole is greater than the sum of its parts” - Aristotle • “The whole is less than the sum of its parts” - Edward Lewis

18. The secondary polytope • For a given genotype space, what fitness shapes are there? • The answer to this parametric fitness shape problem is encoded in the secondary polytope. • For example: • The 2-cube has 2 triangulations. • The 3-cube has 74 triangulations, but only six combinatorial types. • The 4-cube has 87,959,448 triangulations and 235,277 symmetry types.

19. The 74 shapes of fitness landscapes on 3 loci

20. A biallelic three-locus system in HIV • HIV protease: L90M; RT: M184V and T215Y. • Fitness measured in single replication cycle, 288 data points (Segal et al., 2004; Bonhoeffer et al., 2004). • Conditional epistasis:

21. A biallelic three-locus system in HIV

22. 2 7 10 26 32 > 60% In these five shapes, both 001 and 010 are “sliced off” by the triangulations, i.e., the fittest populations avoid the single mutants {M184V} and {T215Y}.Hence we consider 000, 011, 100, 101, 110, 111: HIV random fitness landscape 74 = # (triang. 3-cube)

23. HIV secondary polytope This is the shape of the HIV fitness landscape on PRO 90 / RT 184 / RT 215