# Effect on fitness of mutations

What is the distribution/probability density function (PDF) of impacts on fitness of new mutations?

I very welcome any partial answer that does not give the whole PDF but just some information about the expected value or the variance of this distribution. Information of the kind: "If we consider only beneficial mutations, then the PDF is $P(X=x) = f(x)$" are also welcome.

When I say mutations, one might have want to reduce the concept of mutations only to indels and point mutations.

Of course the answer will depend on the species under consideration and from population to population but I welcome any answer that could give some insights. Eventually, some information according to what is generally assumed to be the PDF of effects on fitness of new mutations might be useful.

Here is a related question

Here is an article that assumes an exponential distribution of effects on fitness of beneficial mutations.

• the PDF should be a conditional one. – WYSIWYG Mar 31 '14 at 11:03
• also your question is quite broad and needs rational modeling for a reasonably accurate answer. – WYSIWYG Mar 31 '14 at 11:19
• @WYSIWYG The PDF should be conditional on what? On the species. Yes obviously. But any information would be welcome. I re-edited my question to make clear what I understand by mutation. I don't feel my question is too broad. Please let me know if you can think of a way of how I could narrow down my question. – Remi.b Mar 31 '14 at 11:49
• you can narrow down by perhaps assuming a hypothetical statistical model – WYSIWYG Mar 31 '14 at 11:58

To a good first approximation $\overline{\Delta f} = 0$. Where $\overline{\Delta f}$ is the mean change in fitness down to any point or indel mutation. The reasons for this are as follows:

1. In the genome of higher organisms, most of the genome is non-functional ("junk") so most mutations will not have any effect regardless of the change made.
2. A substantial proportion of in-frame point mutations will be synonymous mutations that result in the same amino acid being coded for (actually this can have an effect on protein expression but I don't believe anyone has - yet - shown a fitness difference?)
3. Even where a mutation does alter an amino acid many amino acid changes have no measurable effect on the protein produced. Especially where the new amino acid has similar properties to the one it has replaced.
4. Even when a mutation does alter the protein function, or render the product non-functional, in many cases this will not impact fitness since fitness is conditional on the environment in which it is measured and not all genes impact all environments.

So the distribution, whatever it is, will have a large spike at 0. Probably this spike is several orders of magnitude higher than the next highest value. Further, we can be reasonably certain that $\overline{\Delta f} < 0$ since there are more possible ways to break a Gene through a point or indel mutation than there are to improve it. If this is the case, so that there is a large excess of mutations with a negative effect on fitness, we can conclude that the distribution of mutational changes in fitness will be non-normal (negative skew and spike at zero), and that the normal distribution will be a poor approximation.

• I like your reasoning, but dont understand what you mean with the last sentence. – fileunderwater Mar 31 '14 at 14:54
• Reminds me of Pareto optimization. But effect of each mutation on a gene still has to be sampled. – WYSIWYG Mar 31 '14 at 15:04
• @fileunderwater: If it's left skewed and has a central spike at an exact value it therefore will be poorly approximated by any take on the normal distribution. – Jack Aidley Mar 31 '14 at 15:07
• @JackAidley Gotcha. Suggested clarification. – fileunderwater Mar 31 '14 at 22:24
• Take a look at this paper. This talks of a concept similar to Jack Aidley's approach. – WYSIWYG Apr 1 '14 at 4:02

[This is purely speculative]

Assumptions:

• impact on fitness is measured by survival chance
• impact is because of protein coding genes

Probability of a mutation at position $i$

$P(m=i\ |\ g)$

where $g$ is the genome with its annotations.

Probability that activity of some protein changes by X-fold given mutation at $i^{th}$ position(s) in the genome:

$\sum\limits_{i}\sum\limits_{gene} P(a_{gene}= X\ |\ m=i)$

sum over all genes that occupy that locus.

Probability that on a given selection pressure $S$ and protein activity change $X$, the individual survives the selection.

$\sum\limits_{i}\sum\limits_{gene}P(1\ |\ a_{gene}=X,S)$

I am not an expert at Bayesian modeling. But I guess this is one way to go about obtaining the PDF. Each step has to be expanded and combined to get an equation for PDF.

I am not sure if this is of any help at all.

• Thanks a lot for your answer. While it is probably interesting I have to confess that I don't understand everything. I don't really understand why defining a locus as a long enough sequence so that it occupies several genes. I don't think it really addresses the question. If $S$ is the random variable of effect that a mutation has, I am asking about the distribution of $S$: $P(S=s)$. Or as you stated, that might under some assumptions, correspond to $P(a_{gene} = x | m=i)$ but there is no need to sum over all mutations in the genome. I am not asking about the effect of the sum of all mutations – Remi.b Mar 31 '14 at 12:25
• I don't think a purely theoretical model might give much insights (or only a model considering the complex network of protein interactions) but I could be wrong. – Remi.b Mar 31 '14 at 12:26
• By several genes I meant splice variants that occupy the same genomic locus. Only by considering all the positions in the genome will we be able to obtain a PDF. And in this case the effect of a mutation is denoted by the random variable $X$, whereas $S$ is the exerted selection pressure. I totally agree that a purely theoretical model wont give much insights. In this case the prior probabilities have to be calculated from experimental observations. But with a model we can make better predictions. – WYSIWYG Mar 31 '14 at 12:55