# What model of epistasis best fit the reality?

In theoretical population genetics, it is very common to have to assume a model of epistatic interaction. The two most common models are the additive model and the multiplicative model.

Under the additive model the fitness $w$ of an individual carrying all $n$ mutations of effects $[s_1,s_2,...,s_n]$ is

$$w=1-\sum_{i=1}^n s_i$$

, which becomes $w=1-sn$, for the special case when $s=s_i \forall i \in [1,n]$

Multiplicative model

Under the multiplicative model the fitness $w$ of an individual carrying all $n$ mutations of effects $[s_1,s_2,...,s_n]$ is

$$w=\prod_{i=1}^n (1-s_i)$$

, which becomes $w=(1-s)^n$, for the special case when $s=s_i \forall i \in [1,n]$

Old observation

In Population genetics: a concise guide, Gillespie reused data from Mukai (1968) to plot the following graph , where the additive $\left(w=1-sn\right)$ and multiplicative $\left(w=(1-s)^n \right)$ models correspond to the two lines that poorly fit the data. The line that best fit the data is a quadratic model of synergistic epistasis

$$w=1-sn-an^2$$

In all cases, $s$ and $a$ where estimated from these data. We probably have better data and better studies today.

Question

What model of epistatic interaction would you recommend?

Here are some clarifications if needed

Consider $n$ mutations. The fitness of an individual carrying only the first mutation is $1-s_1$. The fitness of an individual carrying only the second mutation is $1-s_2$, etc... the fitness of an individual carrying the $n^{th}$ mutation is $1-s_n$. What is the fitness of the individual carrying all $n$ mutations? Of course, there is no single answer to this question and it will depend on the epistatic interactions between these loci. However, there is probably a general trend that a theoretician would like to consider in its assumptions.

• All models are wrong but some are useful :P – WYSIWYG May 17 '16 at 8:10
• I think that if the genes are actually interacting then they are not independent. If they are not independent, their effects cannot be additive. I have not really studied or worked with the models that you mention above but this is a general sort of rule that additive effects are usually because of independence. – WYSIWYG May 17 '16 at 8:13
• If you're fitting real data, try both; if you're doing a purely theoretical model, use whichever is more mathematically convenient. Note that you haven't defined what you mean by "fitness" here; additive effects on growth rate are multiplicative effects on offspring number. – Daniel Weissman May 18 '16 at 20:43
• I don't understand what is unclear about my definition of fitness. Fitness is the expected number of offspring of an individual. The statistic I use here $w$ is the fitness of an individual relative to a hypothetical individual that bear absolutely no deleterious mutations (and I ignore the existence of beneficial mutations). I do not talk about the growth rate of a population and would be happy to assume that the carrying capacity is not influenced by the population mean fitness. Thanks – Remi.b May 18 '16 at 20:47