### Fitness landscapes and incomplete data

#### Alex Gavryushkin

12 February 2018

Throughout, we consider $n$ biallelic loci, for different $n$.

That is, the set of genotypes is $\mathcal G = \{0,1\}^{n}$.

A fitness landscape is a function $w:\mathcal G \to \mathbb R^+$.

For $g \in \mathcal G$,  $w(g)$ is called the fitness of genotype $g$ and denoted $w_g$.

### Epistasis

is defined as the deviation from the additive expectation of allelic effects: $$u_{11} = w_{00} + w_{11} - (w_{01} + w_{10})$$

Image: Wikipedia

#### Mutation fitness graph

Ogbunugafor et al. Malar. J. 2016

#### Rank orders. The simplest case.

$\small u_{11} = w_{00} + w_{11} - (w_{01} + w_{10})$

### Characterization of epistatic rank orders

Theorem 1. Consider a biallelic $n$-locus system. The number of rank orders which imply $n$-way epistasis is: $\frac{(2^n)! \times 2}{2^{n-1}+1}$

Corollary. The fraction of rank orders that imply $n$-way epistasis among all rank orders is: $\frac{2}{2^{n-1}+1}$

#### Connection between rank orders and mutation graphs

Observation: A partial fitness order implies epistasis if and only if all its total extensions do.

Let $f$ be a linear form with integer coefficients. Assume that the sum of the coefficients of $f$ is zero. A partial fitness order of genotypes $\prec$ implies positive $f$-interaction if $f(w) > 0$ whenever $w$ satisfy $\prec$.

Theorem: A partial order $P = (G, \prec)$ implies positive $f$-interaction if and only if there exists a partition of the set of all genotypes $G$ into pairs $(p_i, n_j)$ such that $p_i \succ n_j$ for all $i, j$, where $p_i$ have positive coefficients in $f$ and $n_j$ negative.

Proof: Exercise.

Corollary: It is polynomial in the number of genotypes ($n^{5/2}$) to check whether a partial order implies $f$-interaction.

### Applications

• HIV-1

• Antibiotic resistance

• Gut microbiome

• Synthetic lethality

• Knockdown cell lines

Methodologically, this allows us to advise further measurements (experiments) for incomplete data sets, thus reducing the number of potential experiments significantly.

#### Example: antibiotic resistance

Mira et al. PLOS ONE, 2015

#### Example: antibiotic resistance

Mira et al. PLOS ONE, 2015

$u_{111} = w_{000} + w_{011} + w_{101} + w_{110} - (w_{001} + w_{010} + w_{100} + w_{111})$

### Understanding three-way interactions

Total three-way interaction?

$\small u_{111} = w_{000} + w_{011} + w_{101} + w_{110} - (w_{001} + w_{010} + w_{100} + w_{111})$

Marginal epistasis?

$\small u_{\color{blue}{0}11} = w_{\color{blue}{0}00} + w_{\color{blue}{1}00} + w_{\color{blue}{0}11} + w_{\color{blue}{1}11} − (w_{\color{blue}{0}01} + w_{\color{blue}{1}01}) − (w_{\color{blue}{0}10} + w_{\color{blue}{1}10})$

Conditional epistasis?

$\small e = w_{\color{blue}{0}00} − w_{\color{blue}{0}01} − w_{\color{blue}{0}10} + w_{\color{blue}{0}11}$

Total mess!

### Algebraic Geometry sorts out the mess!

$e = \frac12(u_{011} + u_{111})$

In general, the four interaction coordinates $$u_{011}, u_{101}, u_{110}, u_{111}$$ allow to describe all possible kinds of interaction!

There are 20 types of three-way interaction and they are the circuits of the three-cube.

Yep, we've got the list!

\scriptsize \begin{align*} a&= w_{000}-w_{010}-w_{100}+w_{110} & m&=w_{001}+w_{010}+w_{100}-w_{111}-2w_{000}\\ b&=w_{001}-w_{011}-w_{101}+w_{111} & n&=w_{011}+w_{101}+w_{110}-w_{000}-2w_{111}\\ c&=w_{000}-w_{001}-w_{100}+w_{101} & o&=w_{010}+w_{100}+w_{111}-w_{001}-2w_{110}\\ d&=w_{010}-w_{011}-w_{110}+w_{111} & p&=w_{000}+w_{011}+w_{101}-w_{110}-2w_{001}\\ e&=w_{000}-w_{001}-w_{010}+w_{011} & q&=w_{001}+w_{100}+ w_{111}-w_{010}-2w_{101}\\ f&=w_{100}-w_{101}-w_{110}+w_{111} & r&=w_{000}+w_{011}+ w_{110}-w_{101}-2w_{010}\\ g&=w_{000}-w_{011}-w_{100}+w_{111} & s&=w_{000}+w_{101}+ w_{110}-w_{011}-2w_{100}\\ h&=w_{001}-w_{010}-w_{101}+w_{110} & t&=w_{001}+w_{010}+w_{111}-w_{100}-2w_{011}\\ i&=w_{000}-w_{010}-w_{101}+w_{111}\\ j&=w_{001}-w_{011}-w_{100}+w_{110}\\ k&=w_{000}-w_{001}-w_{110}+w_{111}\\ l&=w_{010}-w_{011}-w_{100}+w_{101}\\ \end{align*}
\scriptsize \begin{align*} a&= w_{000}-w_{010}-w_{100}+w_{110} & m&=w_{001}+w_{010}+w_{100}-w_{111}-2w_{000}\\ b&=w_{001}-w_{011}-w_{101}+w_{111} & n&=w_{011}+w_{101}+w_{110}-w_{000}-2w_{111}\\ c&=w_{000}-w_{001}-w_{100}+w_{101} & o&=w_{010}+w_{100}+w_{111}-w_{001}-2w_{110}\\ d&=w_{010}-w_{011}-w_{110}+w_{111} & p&=w_{000}+w_{011}+w_{101}-w_{110}-2w_{001}\\ e&=w_{000}-w_{001}-w_{010}+w_{011} & q&=w_{001}+w_{100}+ w_{111}-w_{010}-2w_{101}\\ f&=w_{100}-w_{101}-w_{110}+w_{111} & r&=w_{000}+w_{011}+ w_{110}-w_{101}-2w_{010}\\ \color{blue}{g}&\hskip{2pt}\color{blue}{=w_{000}-w_{011}-w_{100}+w_{111}} & s&=w_{000}+w_{101}+ w_{110}-w_{011}-2w_{100}\\ h&=w_{001}-w_{010}-w_{101}+w_{110} & t&=w_{001}+w_{010}+w_{111}-w_{100}-2w_{011}\\ i&=w_{000}-w_{010}-w_{101}+w_{111}\\ j&=w_{001}-w_{011}-w_{100}+w_{110}\\ k&=w_{000}-w_{001}-w_{110}+w_{111}\\ l&=w_{010}-w_{011}-w_{100}+w_{101}\\ \end{align*}