Epistasis and Shapes of Fitness Landscapes Niko Beerenwinkel, Lior Pachter, Bernd Sturmfels Department of Mathematics University of California at Berkeley.

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

“The whole is greater than the sum of its parts” - Aristotle

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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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Example:

00 11

10

01

01

0011

10

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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 1:

One circuit: 000

001

010

100

111

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Example 2:

Four circuits:

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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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Example 3: Towards the human genotope

HapMap consortium, 2005

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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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The 74 shapes of fitness landscapes on 3 loci

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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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A biallelic three-locus system in HIV

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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)

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HIV secondary polytope

This is the shape of the HIV fitness landscape on PRO 90 / RT 184 / RT 215