Gene interactions:
a geometric approach
Alex Gavryushkin
Joint work with:
- Kristina Crona, American U., Washington, DC, USA
- Bernd Sturmfels, U. of California, Berkeley, USA
- Ewa Szczurek, U. of Warsaw, Poland
- Niko Beerenwinkel, ETH Zurich, Basel, Switzerland
December 1, 2016
Throughout, we consider $n$ biallelic loci, for various values of $n$
Notation
For a genotype $g$, $w_g$ denotes the fitness of $g$,
for example, $w_{11}$ is the fitness of the double mutant.
Epistasis $u_{11}$
Additive allelic effect $=$ no epistasis:
$$
w_{11} + w_{00} = w_{01} + w_{10}
$$
Deviation from the additive expectation of allelic effects:
$u_{11} = w_{00} + w_{11} - (w_{01} + w_{10})$
Understanding three-way interactions
Marginal epistasis?
$\small u_{\color{blue}{0}11} = w_{\color{blue}{0}00} + w_{\color{blue}{1}00} + w_{\color{blue}{0}11} + w_{\color{blue}{1}11} − (w_{\color{blue}{0}01} + w_{\color{blue}{1}01}) − (w_{\color{blue}{0}10} + w_{\color{blue}{1}10})$
Total three-way interaction?
$\small u_{111} = w_{000} + w_{011} + w_{101} + w_{110} - (w_{001} + w_{010} + w_{100} + w_{111})$
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!
$e = u_{011} + u_{111}$
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 interaction and they are known as circuits to Algebraic Geometry 111 students
Yep, we've got the list!
$$
\scriptsize
\begin{align*}
a&= w_{000}-w_{010}-w_{100}+w_{110} & m&=w_{001}+w_{010}+w_{100}-w_{111}-2w_{000}\\
b&=w_{001}-w_{011}-w_{101}+w_{111} & n&=w_{011}+w_{101}+w_{110}-w_{000}-2w_{111}\\
c&=w_{000}-w_{001}-w_{100}+w_{101} & o&=w_{010}+w_{100}+w_{111}-w_{001}-2w_{110}\\
d&=w_{010}-w_{011}-w_{110}+w_{111} & p&=w_{000}+w_{011}+w_{101}-w_{110}-2w_{001}\\
e&=w_{000}-w_{001}-w_{010}+w_{011} & q&=w_{001}+w_{100}+ w_{111}-w_{010}-2w_{101}\\
f&=w_{100}-w_{101}-w_{110}+w_{111} & r&=w_{000}+w_{011}+ w_{110}-w_{101}-2w_{010}\\
g&=w_{000}-w_{011}-w_{100}+w_{111} & s&=w_{000}+w_{101}+ w_{110}-w_{011}-2w_{100}\\
h&=w_{001}-w_{010}-w_{101}+w_{110} & t&=w_{001}+w_{010}+w_{111}-w_{100}-2w_{011}\\
i&=w_{000}-w_{010}-w_{101}+w_{111}\\
j&=w_{001}-w_{011}-w_{100}+w_{110}\\
k&=w_{000}-w_{001}-w_{110}+w_{111}\\
l&=w_{010}-w_{011}-w_{100}+w_{101}\\
\end{align*}
$$
$$
\scriptsize
\begin{align*}
a&= w_{000}-w_{010}-w_{100}+w_{110} & m&=w_{001}+w_{010}+w_{100}-w_{111}-2w_{000}\\
b&=w_{001}-w_{011}-w_{101}+w_{111} & n&=w_{011}+w_{101}+w_{110}-w_{000}-2w_{111}\\
c&=w_{000}-w_{001}-w_{100}+w_{101} & o&=w_{010}+w_{100}+w_{111}-w_{001}-2w_{110}\\
d&=w_{010}-w_{011}-w_{110}+w_{111} & p&=w_{000}+w_{011}+w_{101}-w_{110}-2w_{001}\\
e&=w_{000}-w_{001}-w_{010}+w_{011} & q&=w_{001}+w_{100}+ w_{111}-w_{010}-2w_{101}\\
f&=w_{100}-w_{101}-w_{110}+w_{111} & r&=w_{000}+w_{011}+ w_{110}-w_{101}-2w_{010}\\
\color{blue}{g}&\hskip{2pt}\color{blue}{=w_{000}-w_{011}-w_{100}+w_{111}} & s&=w_{000}+w_{101}+ w_{110}-w_{011}-2w_{100}\\
h&=w_{001}-w_{010}-w_{101}+w_{110} & t&=w_{001}+w_{010}+w_{111}-w_{100}-2w_{011}\\
i&=w_{000}-w_{010}-w_{101}+w_{111}\\
j&=w_{001}-w_{011}-w_{100}+w_{110}\\
k&=w_{000}-w_{001}-w_{110}+w_{111}\\
l&=w_{010}-w_{011}-w_{100}+w_{101}\\
\end{align*}
$$
This is known as Beerenwinkel-Pachter-Sturmfels approach,
which provides a complete picture of interactions!
BUT
the approach is
Hence, we come to two research questions
Problem 1: What if no (credible) fitness measurements are available?
Like in this malaria drug resistance data set:
Ogbunugafor et al. Malar. J. 2016
Results at a glance
- We provide a complete characterization of fitness graphs that imply circuit interaction (think epistasis).
- Fitness graphs arise in competition-like experiments and include:
- Rank orders
- Mutation graphs
(Preprint(s) with Crona, Beerenwinkel, and others to appear)
Rank orders. The simplest case.
$\small u_{11} = w_{00} + w_{11} - (w_{01} + w_{10})$
Characterization of epistatic rank orders
Theorem. 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:
\[
\frac{2}{2^{n-1}+1}
\]
Mutation graph
Connection between rank orders and mutation graphs
Applications
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
Example: antibiotic resistance
Example: antibiotic resistance
Mutation graph
Mutation graph
Mutation graph
Mutation graph
Mutation graph
Results in more detail
Efficient methods for:
- Circuit interaction inference (including epistasis and three-way interaction) for total orders
- Complete analysis of partial orders (including mutation graphs) with "distance to interaction" inference
- Suggestions for possible completions in case of missing data and/or high uncertainty
Software (pre-release stage):
https://github.com/gavruskin/fitlands
Problem 2: What if the number of genes (loci) is 20,822?
- 2^20822 of conditional epistases?
- 2^20822 measurements to estimate marginal epistasis?
Not in this life
Concrete example: genome-wide RNAi perturbation screens
20,822 genes, 90,000 "trials" (siRNA's)
RNAi perturbation screen
Two ways out
- Isolate a small number of "interesting" genes, e.g. main fitness drivers (like we did in the HIV study)
- Add statistical assumptions, for example:
(Ongoing work with Schmich, Szczurek, Beerenwinkel, et al.)
Want to learn more?
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Thanks for your attention!
and stay tuned