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