Scientists Use Game Theory to Determine the Importance of Individual Genes

Source: mknowles via flickr

How can game theory possibly used in genetics?

That’s what I thought when reading about new genetics research. I can’t say I have a grasp on the biology, but after some investigation, I now have a sense of how game theory comes in.

I’ll summarize the news and then consider the role of game theory.

News: scientists show single-gene studies miss the picture

Be weary of headlines that claim scientists have discovered things like the “gene for obesity” or the “gene for intelligence.” These approaches may not be capturing the full picture because of interaction.

The standard research method, reported by Patrick Barry of Science News, is engineering organisms by “knocking out” a single gene and studying the change. The logic is as follows: if a trait does not develop after a single-gene knockout, one could conclude the gene plays a role in the trait.

While single-gene studies are important, there are two obvious limitations to them. First, many traits are the expression of multiple genes. Single-gene knockouts will miss the role of each individual gene. Second, genes could play redundant roles as a type of biological security. Single-gene knockouts will entirely miss the role of the redundant genes because others compensate during the experiment.

New research quantifies how big of a problem this is. Scientists from Israel and Germany have shown that single-gene studies miss at least 33 percent of the genes important for yeast growth. What’s the proposed answer? It is using multiple-gene knockouts to get a better sense of the interaction.

The problem then becomes one of too much information: if multiple genes are important, and some are possibly redundant, how can you quantify the importance of any particular gene?

And this is where game theory comes in. The answer is to reformulate the question in terms of a game. Think about each gene as a “player” and each result as an “outcome.” In this language, the question becomes determining the importance of a particular player to a cooperative outcome. Solutions to this question have been around since 1950.

I am not sure how the geneticists approached the issue, but I do have an idea of the process. I want to show how one can determine power of players using a classic game theory problem that has two redundant workers.

The Glove Game

Imagine there are three workers in a factory. Suppose two workers can only make left-hand gloves while a third can only make right-hand gloves.

Consumers only care about pairs of gloves, so let’s say the owner only cares about producing pairs of gloves–one left-hand and one-right hand together. That is, unpaired gloves serve no purpose. Under these assumptions, how important is each player to the final product?

Intuition indicates that the right-hand glove maker is very important. If that worker calls in sick, or is on strike, then it’s impossible to create the product.

Furthermore, the two workers making the left-hand gloves are individually less important because they perform redundant tasks. If one of them were sick, or on strike, the other might cover.

Nonetheless, the two left-hand glove makers are definitely important as a group. If both called in sick, or both protested, it would be impossible to create the product.

How should the owner fairly assess the value of each worker? How should the workers be paid relative to each other?

The owner needs a tool to compute each worker’s contribution to the whole. Game theory offers several methods, and one common method is the Shapley value. This is the idea that was later applied to create the Shapley-Shubik power index, which I discussed last week with voting power.

The Shapley value is very similar to the Shapley-Shubik power index, but I’ll recap the idea for completeness.

The Shapley Value

• Informally speaking, the Shapley value for a player is the marginal contribution to a group, when considering all possible orders the group would form.

For the glove game, the Shapley value has the following interpretation. Assume that production takes place sequentially, one player at a time, and in a random order. The Shapley value is the probability a player completes the glove pair, when considering all possible orders of production.

Solution: The Glove Game

Let’s call the players 1L, 2L, 3R for the workers that can make gloves left, left and right.

How important is each player to creating a product?

There are 6 possible production line orders:

1L 2L 3R
1L 3R 2L
2L 1L 3R
2L 3R 1L
3R 1L 2L
3R 2L 1L

The first player that completes the glove is marked in bolded. For instance, in the first sequence, players 1 and 2 both make left-hand gloves, so it is the third player that completes the product and is so marked in bold.

Note that player 3, the one making the right glove, completes the product in 4 of the 6 cases. This means player 3 has a Shapley value of 2/3.

The players making the left glove are less important as each completes the product in one sequence apiece. This means each of them has a Shapley value of 1/6.

The economic interpretation is that players 1 and 2 contribute 1/6 to the product while player 3 contributes 2/3.

The Shapley values can be used as the ratios to distribute profits. If the glove produces $600 of profit, then the fair division would be $600 x 1/6 = $100 to players 1 and 2 and $600 x 2/3 = $400 to player 3.

My guess of how single-gene knockouts would miss the picture

Imagine instead of workers and gloves, the game were about genes and traits. Think about each player as a “gene” and the glove as the expression of a “trait.”

The glove (“trait”) would still be created even if either of the players 1 or 2 (“redundant genes”) were experimentally knocked out. Nonetheless, it is clear the glove depends on at least one of the redundancies to work. This is something a single-gene study would miss.

Now, I caution taking this analogy too literally because I lack the expertise to verify precisely what the biology researchers are doing (please share in the comments if you do).

Furthermore, I would guess gene interaction and trait expression are more complicated production processes, and they might even depend on exogenous factors. Nevertheless, this application of computational game theory appears to be a significant improvement over the single-gene knockout method. I look forward to more work in this area.

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