epistasis and shapes of fitness landscapes niko beerenwinkel, lior pachter, bernd sturmfels...
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Epistasis and Shapes of Fitness Landscapes
Niko Beerenwinkel, Lior Pachter, Bernd Sturmfels
Department of Mathematics
University of California at Berkeley
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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
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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
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Epistasis
Two-locus two-alleles: ab aB Ab ABwith fitness landscape wab waB wAb wAB
aB
Ab
fitne
ss
genotype
ab
AB?
AB?
AB?
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Epistasis
Two-locus two-alleles: ab aB Ab ABwith fitness landscape wab waB wAb wAB
fitne
ss
genotype
aB AB
Abab
wab+wAB = wAb+waB
wab+wAB > wAb+waBpositiveepistasis
wab+wAB < wAb+waBnegativeepistasis
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Geometric perspective
Two-locus two-alleles: 00 01 10 11with fitness landscape w00 w01 w10 w11
epistasis u = w00 + w11 – w01 – w10
u = 0 u > 0u < 0
Two generic shapes of fitness landscapes
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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
Populations and the genotope
population simplex
marginalization map
allele frequency space
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A fitness landscape is a function . Linear functions have no interactions, so consider the
interaction 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.
Fitness landscapes and interactions
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Example 3: The vertebrate genotopes
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Margulies et al., 2006.
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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
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Fittest populations with fixed allele frequency
u = 0 u > 0u < 0
{00, 01, 10}{01, 10, 11}
{00, 01, 10, 11} {00, 01, 11}{00, 10, 11}
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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
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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.
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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:
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HIV random fitness landscape
> 60%
2 7 10 26 32
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:
74 = # (triang. 3-cube)