# Mutation-Selection Equilibrium

How do you permanently increase or decrease the fitness of a population (in terms of population genetics). Is it better to intensify the strength of selection so that deleterious mutations are removed, or relax the strength of selection so that mutations are better tolerated. This is something that is relevant to the current human condition, where there is an increased amount of medical intervention and (in the very recent past) a relaxation of some adaptive demands in the environment (this will be misunderstood, human culture has placed additional demands upon humans in the longer timescale, but anyway...). Arguments can be made for either scenario and---it turns out that it really doesn't matter. Strangely enough the strength of selection cannot change the average fitness (at equilibrium) of a population.

For this post only think of mutations as deleterious (they lower an organism's fitness) if they have any effect at all. Most mutations either have no fitness effect (are selectively neutral) or lower fitness. Very few mutations in the genome would actually increase the fitness of an organism (adaptive).  Again think about making random changes to a car. Most changes, slightly reducing the length of the radio antenna, either make essentially no difference or, reroute the fuel line to the windshield wiper pump, are a very bad idea in terms of performance of the car. It would be very rare to make random changes that increase the performance of the car.

So, we know there are deleterious mutations that exist in a population that can result in what we identify in humans as a genetic disease. Many of these are recessive such as cystic fibrosis or Tay-Sachs disease; however, some are dominant such as Huntington's disease or Marfan's syndrome. And some do not fit this simple category such as X-linked hemophilia. Why do human and other species have so many deleterious alleles in the population?

The alleles are generated by mutation and removed by selection. We can have a mutation rate  and a strength of selection (relative to unmutated alleles) acting on those mutations . If we pretended the alleles acted independently, that pairing them together into diploid genotypes did not matter, then we can easily write down the equilibrium allele frequency, , of the mutation.



This is the rate that the allele is generated, , out of the total rate of input by mutation and removal by selection. For example, if the strength of selection against a mutant is 20% (a 20% fitness reduction relative to individuals with unmutated alleles) and the mutation rate is 0.1% then the equilibrium allele frequency is approximately one half of a percent, 0.5%. On the other hand if the fitness reduction is only 1% the allele can reach a higher frequency in the population at the same mutation rate (=0.1%) the equilibrium becomes approximately 9%, which is a dramatically high frequency for a dominant deleterious allele. (This is mathematically equivalent to bi-directional mutation; however, here the "back mutation" to restore the original state is selection.)

Now lets keep track of genotypes and the case where the fitness effect of the allele is dominant (with some simplifying assumptions). If it is dominant and deleterious it is probably rare (not like the 9% case above), so homozygotes,  are exceedingly rare and can be safely ignored because they contribute very little to the equilibrium dynamics. Mutations occur on the non-mutant allele copies  (regardless if they are present in a homozygote or heterozygote). Selection removes a portion of heterozygotes, according to Hardy-Weinberg heterozygotes are expected to appear at a frequency of  and an  fraction of these are removed so the genotype rate of removal is  .

(, the relative fitness to the unmutated homozygote is  which adjusts the frequency of the heterozygotes  due to selection.)

Only half of the alleles in the heterozygotes are the mutant allele so the rate of removal of the mutant allele is . (Here we are focused on the case where the mutant allele is rare, which allows us to ignore  in the first place. So, . The frequency of heterozygotes is then approximately . However, we want to keep track of the change in  due to  so we divide  by 2. Selection is also removing non-mutant alleles in heterozygotes but the non-mutant homozygotes are much more common so we just ignore the other half of the alleles that are removed due to selection; this comes from the assumptions of large population size and rare mutant allele frequency.)

At equilibrium the rate of input and removal of the mutation are equal.



Rearrange and simplify.





Using the two examples above with a mutation rate of  the equilibrium allele frequency is predicted to be 0.5% for a fitness reduction of 20% and 10% for a fitness reduction of 1%, which is almost equivalent to the case where alleles are acted on independently (as haploids). (If  then .)

What if the fitness effect is recessive? Then selection only removes the mutant homozygotes which are predicted to occur at a frequency of . In this case selection can proceed more efficiently, in a sense, and mutant alleles are removed in pairs. All of the alleles in the homozygote are the mutant form so there is no adjustment necessary like dividing by two in the heterozygotes.



To keep this from getting messy let's use the  trick now (which is actually less appropriate in this case because higher frequencies can be attained, but for now still assume that mutant alleles are very rare, , which is true for many mutations that result in human genetic diseases).







Using our two examples again, with a mutation rate of  the equilibrium allele frequency is predicted to be 7% for a fitness reduction of 20% and 32% for a fitness reduction of 1%. Now the equilibrium allele frequencies are much higher, when very rare (when ) the difference can be many orders of magnitude. Masking the fitness effects in heterozygotes (carriers) allows the allele to get to unexpectedly high equilibrium frequencies.

Now let's look at the average effect in the population. The average fitness  in a population can be calculated as the fitness of each genotype multiplied by its corresponding frequency. Let's say the fitness of unmutated homozygotes is 100% or 1. In the case of a dominant fitness effect



Expand and simplify



Let's say 



Substituting in 



The average fitness is one minus twice the mutation rate.

On the other hand if selection is recessive



Expand and simplify



Substituting in 



The average fitness is one minus the mutation rate.

Interestingly, the average fitness in the population does not depend on the fitness of the genotypes within the population; it is only a function of the mutation rate! (However, it does depend on the type of dominance; recessive deleterious effects result in both a higher equilibrium allele frequency and a higher average fitness.) This seems like a paradox at first.  In our examples from above, if selection is dominant, then with a mutation rate of 0.1% the average fitness within the population is 99.8% regardless of how strong selection is against the mutation. If the mutation is recessive the average fitness is actually higher 99.9%, again regardless of the strength of selection against the mutation (this is related to selection being more efficient when it acts upon pairs of mutant alleles rather than one at a time in heterozygotes).

Why is average fitness only determined by the mutation rate? The trick to understand this is to realize that the strength of selection against a mutation and the frequency of the mutation in the population at equilibrium cancel out in terms of the average effect in the population. A mutation with a strong effect can only exist at a low frequency and only affect a few individuals, while a mutation with a weak effect attains a higher frequency and affects more individuals. The average effect of these two mutations, that have very different effects on fitness, in a population at equilibrium is the same.

However, going back to the original question of this post, changing the mutation rate can permanently change the average fitness of a population. How do you change the mutation rate? Well certain chemicals and radiation are well known examples. (Also, in a recent article Michael Lynch points out that if relaxed selection affects mutations in genes that affect the mutation rate itself, DNA repair, etc., then there could be a feedback effect where selection does influence equilibrium average fitness by altering the mutation rate.)

To put this in perspective, mutations are not rare unusual events that can safely be ignored; they affect all of us. We all have new mutations that can be passed on to our children. Whole genome sequencing estimates this as high as to be around 40-80 new mutations depending on the age of our father. (Click on the image below for the source information.)

Each year of father's age adds on average two new mutations. The age of the father is also a risk factor for diseases like autism and schizophrenia and new mutations might be playing a role.

Above ground nuclear testing released large amounts of radioactive particles into Earth's atmosphere which spread planet-wide. At the height of the Cold War before the 1963 partial test ban the amount of radiation in the atmosphere was almost doubled (click on the image for a link to more information about how it was generated).

This had very real effects, for example it created a market for pre-war steel for specialized equipment like Geiger counters to test levels of radiation (steel made after WWII was contaminated with atmospheric radiation). A particularly valuable source are WWI battleship wrecks that are protected under water from contact with the atmosphere.

The increase in radiation alarmed some population geneticists like Hermann Muller who studied mutations and their effects on fitness. He helped raise awareness of the issue and his work among others contributed to the partial test ban treaty.

Furthermore, we come into contact with a wide range of industrial chemicals, many of which are seriously toxic and/or some of which can cause heritable mutations.

It is not known how this exposure might translate into increases in mutation rates (and to what degree this might contribute to the rates of genetic diseases). It would be interesting to estimate genome-wide mutation rates, in humans and other species if possible, over the last few centuries by comparing relatives sharing DNA sequences that are connected by different numbers of generations before and after certain points in history to see if there is a measurable effect.

# National Academies of Sciences briefing on gene drive technology

Today the NAS released a report on gene drive technology:

http://nas-sites.org/gene-drives/2016/05/26/report-release/

Excerpts from a New York Times article about the report:

'On Wednesday, the National Academies of Sciences, ... endorsed continued research on the technology, concluding after nearly a yearlong study that while it poses risks, its possible benefits make it crucial to pursue. ... The report underscores that there is not yet enough evidence about the unintended consequences of gene drives to justify the release of an organism that has been engineered to carry one. ... At the same time, it is uncertain how the technology will be regulated. Existing laws, the report noted, are aimed at containing genetically engineered organisms rather than managing those whose purpose is precisely to spread swiftly. ... Coming up with an international regulatory framework is especially crucial, members of the committee said, given that gene drives will not recognize national or political boundaries. For now, the United States Food and Drug Administration has authority over animals that have been engineered with foreign DNA under a rule that regards them as a type of drug. But the report suggests that other agencies, like the Fish and Wildlife Service or the Bureau of Land Management, might be seen to have a stake in the ecological concerns at the heart of gene drive experiments. ... Some independent scientists say the panel, which included ethicists, biologists and others, struck a good balance by permitting more gene drive research while limiting the use of the technology. But opponents of genetic engineering argue that the panel should have demanded a halt to research on gene drives, at least until some of the many questions it raised are answered. ... The committee considered six case studies, including using gene drive to control mice destroying biodiversity on islands, mosquitoes infecting native Hawaiian birds with malaria, and a weed called Palmer amaranth that has become resistant to herbicides and a scourge for some farmers. Each potential use of gene drive carries its own set of risks and benefits, the report says, and should be assessed independently. ... The group recommends “phased testing,’’ which would include safeguards at each step before eventually releasing organisms into the wild, but it also noted the new ethical challenges posed by how to obtain consent from people whose environments might be affected by such a release. “There are few avenues for such participation,” the report noted, “and insufficient guidance on how communities can and should take part.”'

# National Academies of Sciences Report on Genetically Engineered Crops

This was just released yesterday from a committee chaired by Fred Gould:

http://nas-sites.org/ge-crops/2016/04/27/report-release/

"This consensus report examines a range of questions and opinions about the economic, agronomic, health, safety, or other effects of genetically engineered (GE) crops and food. Claims and research that extol both the benefits and risks of GE crops have created a confusing landscape for the public and for policy makers. This report is intended to provide an independent, objective examination of what has been learned since the introduction of GE crops, based on current evidence."

# DARPA, Gene Drive Technologies, and the ENMOD Treaty

I just returned from a "gene drive" workshop at NCSU's Genetic Engineering and Society Center. There is a lot to talk about from the meeting. Here I want to focus on a couple of specific details that came up in reference to the military funding of genetic technology. The meeting was held under Chatham House Rule, so I cannot identify the people who made the original statements. Some of these were in group discussions and some of these were personal one-on-one conversations. As a brief, overly terse, background statement to describe a complex field: gene drive technology is a new emerging technology that is potentially very powerful and could be used for beneficial humanitarian and species conservation applications where other methods have fallen short in their long term effectiveness.

First of all, I was told that DARPA is interested in funding gene drive technology for environmental modifications. DARPA helps to develop new technologies for military applications.  This could be for both species conservation applications as well as preventing infectious disease (and also there is obviously the possibility of malicious hostile use in military applications but this was not brought up). Apparently a man named Dr. Jack Newman (link) is slated to become the program manager of mosquito gene drive technology at DARPA.

So---to be frank---I believe this is potentially a very bad idea for many reasons. The first is strategic. If these kinds of technologies are to ultimately be used for beneficial reasons they must be acceptable in some degree to the public so that they can become adopted and utilized. The Pacific Islands have a very negative track record of being used for testing grounds of new technologies. This ranges from classical bio-control releases of invasive species, to loss of traditional land to military activities, to, probably the most glaring example, nuclear testing in the Marshall Islands that displaced Native People and resulted in a region becoming uninhabitable from the resulting radiation (also note French nuclear testing, under protest, in Tureia, link). There is nested within the issues of the loss of self determination resulting from colonialism by many Western Countries across the Pacific. Like it or not, public perception is a very real force that cannot be ignored. The Three Mile Island accident in 1979 led to an effective moratorium on new nuclear reactor construction until 2012; however, many of these new projects have also been canceled with the more recent Fukushima disaster also playing a role. The public reaction to GM Crops has also had a very real effect on the laws surrounding the technology and adoption of the technology around the world including in the Pacific (e.g., the GM Taro and GM Papaya controversies in Hawai'i, link).* Right or wrong, in the Pacific, military funding of a new technology will be initially evaluated within the perspective of other military tests of new technologies and the effects this has had on the people of the Pacific Islands. Even more relevant to gene drive technologies, in the 1970's a World Health Organization project to test the release of sterile mosquitoes in India (to suppress the local population and limit the transmission of disease to humans) was shut down due to public perceptions that it might also be a secret military bio-warfare test (link, incidentally there are also some documents on WikiLeaks related to this).

In a broader ethical-moral sense (and this is very much a personal opinion from the perspective of a US citizen) are we comfortable with the military guiding and controlling the research that goes on in our country? This may sound like hyperbole; however, a comparison of the huge difference in the levels of US military funding (on the order of $610 billion) and National Science Foundation funding (on the order of$7 billion) is objectively dramatic. Advances in research depend on grant funding and support. Which technologies government funding agencies choose to support affects not only the advancement of these technologies but the direction they develop in and as a direct result the future applications of these technologies (the history of Project Orion is one example where limited funding sources and issues of potential military uses caused development focused on military applications yet ultimately stopped a line of scientifically promising yet controversial research, link).

Ideally, for gene drives technologies to be able to realize their potential in beneficial applications, they should be supported and developed by sources other than the military and private companies---and yes, this is strongly motivated by public perception as well as ethical principles. Scientific funding bodies as well as state and local funding have more of a long term potential benefit than is initially apparent. Furthermore, accepting funding from the military lends false support to continuing the objectively inflated funding of the military at the expense of government agencies devoted to scientific research (NSF and others); at the end of the day the military can say that it should continue to receive research funding because of the projects it has supported, but this comes with a social cost. Wouldn't it be better if NSF could make this statement instead without the social cost?

Okay, now comes the ace card that I have been hiding so far in this article... At the meeting, someone brought up (within the context of more "traditional" synthetic biology) that the US is a signatory to the ENMOD international treaty which came into force October 5, 1978. This treaty prohibits the military from using environmental modification technologies that have widespread and/or long-lasting effects. Interestingly, "Environmental Modification Technique includes any technique for changing – through the deliberate manipulation of natural processes – the dynamics, composition or structure of the earth, including its biota" (full text). This gets into philosophical discussions about the role of physical coercion by the military and the state, which I do not want to go into here (links for reference, military, monopoly on violence); however, I will appeal to the common-sense notion that military force by a state is a hostile act although this is more difficult to realize when it is done in a way that aligns with your own interests.  Since gene drive technologies are deliberate manipulations of natural biological processes with long term and possible widespread effects on manipulation of the environment, is DARPA military funding of gene drive technology even legal according to international treaty that the US agreed to support?

# The evolution of antibiotic resistance

Here is one result from this semseter's genetics teaching lab that I wanted to share. The students grew bacteria on a series of gradient media that had increasing concentrations of an antibiotic. At the end of the experiment the bacteria could grow on levels of antibiotic that would have prevented growth before the experiement (which we tested with a control that was genetically identical at the beginning of the experiment and was not exposed to antibiotics). The sucessive generations of bacteria evolved by mutations and selection to tolerate the antibiotic. (One of the goals of this was to show the students an example of evolution in action and illustrate the risks of over-using antibiotics.) We then measuered levels of gene expression for all the genes in the genome and identified which genes had increased their acitvity and which ones had decreased acitivity to allow them to survive (by extracting RNA and hybridizing it to an Affymetrix "GeneChip E. coli Genome 2.0 Array"). Next year I'm planning to have the students sequence some of the genes involved and try to find the precise mutations that have changed gene expression levels.