Epistasis and Shapes of Fitness Landscapes Niko Beerenwinkel, Lior Pachter, Bernd Sturmfels...
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Transcript of Epistasis and Shapes of Fitness Landscapes Niko Beerenwinkel, Lior Pachter, Bernd Sturmfels...
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
5
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)