The paper I discuss here is a review article from 2005 by H. Allen Orr giving a history of the theory of adaptation. I read this paper because I wanted to learn about Fisher’s Geometric Model, this paper does an excellent job of introducing the theory, what purpose it serves, and gives a history of how the field has progressed through time.

First and foremost, let’s clarify that selection is not adaptation, understanding this is fundamental. Adaptation, as Fisher himself put it, is “the progressive increase of fitness of each species of organism” – the net effect is that, over time, species become better suited to their environment. Selection can be the force driving adaptation in the species, but it can also be maladaptive to the species. The example Orr uses is of selfish genetic elements which are spread through a species by selection but are often maladaptive to the species.

We need a theory of adaptation because there are many questions about the process of adaptation that without a theory of adaptation can not be answered. Orr highlights numerous questions which are reliant on such a theory being available; does adaptation comes from novel or standing genetic variation? does adaptation come about by large or small effect mutations? how is adaptation affected by proximity to the adaptive peak? and do complex organisms evolve more slowly than simple ones? Here I discuss Fisher’s Geometric Model in relation to some of these questions to help demonstrate the power of the model.

Fisher’s Geometric Model** **is a model of adaptation proposed in The Genetical Theory of Natural Selection by R. A. Fisher in 1930. This book is a classic piece of literature in evolutionary biology, but it’s also complex and this has lead to some controversy, misunderstanding, and slow uptake of Fisher’s ideas – just look at Fisher’s Fundamental Theorem of Natural Selection for a prime example. Fisher used the geometric model to explain how the process of adaptation depends on the proximity of a population to it’s adaptive peak, the number of traits affecting adaptation, and show that large effect mutations are less likely to result in adaptation. Key to the understanding is remembering that fitness is determined by many characters all at once and that different combinations of values for each character can produce the same fitness.

In this figure we have fitness as a response determined by a number of characters, n. In the blue population fitness is determined by 2 characters, A and B, and each character has two forms A/a and B/b respectively, therefore it can occupy four independent states – the circles. The fittest variant has the character array AB, and the least fit is ab. The two intermediate values are the individuals carry either Ab or aB as there characters, this is two routes to the same fitness value. Moving to the yellow population, where n=3, we have three characters (introducing character C/c to our array of traits determining fitness) which gives more potential combinations of character values. What is apparent from this is that complexity increases both the number of routes and levels of fitness, or adaptedness. Keep this in mind, I’ll be coming back to this, but first I need to cover two more key elements.

The process of adaptation can be graphically represented like this figure, where the yellow dot represents a specific combination of characters and character values, occupying a plane of adaptedness (the dashed circle) which surrounds the adaptive peak (the red dot). All combinations of traits that occupy the same plane as the yellow dot would have the same fitness but by a different array of character values. The yellow dot is a distance, *z*, from the adaptive peak. A mutation, with effect size *r*, may occur which moves the yellow dot to a new position at the end of the blue line. Key points: *z *is the distance between the population’s current plane of adaptedness and the adaptive peak, and *r *is the effect of a new mutation on the adaptedness.

Looking at the first population in this figure, the yellow dot, which has a mutation with effect size *r. *Here the yellow population has an increase in fitness so becomes slightly better adapted – the new position occupied is closer to the optimum. The purple population has a smaller increase along the same trajectory, which is more adaptive than that experienced in the yellow population. It is no stretch of the imagination to suppose a very large effect mutation which goes beyond the farthest boundary of the dashed circle, this would reduce fitness and would not lead to adaptation – the population would not take on the new form and therefore not become better adapted. Effect size of a mutation will influence it’s adaptiveness, again keep this point in mind.

The final key consideration is that of proximity to the adaptive peak. The distance between a current position and the adaptive peak is the value *z. *Compare these two populations, the yellow population is currently far from its adaptive peak, the purple population is very close to its adaptive peak. A mutation arises in both populations with the same effect size, in the yellow population it improves fitness and will spread, in the purple population the mutation overshoots the boundary of the adaptive plane, just like a large effect mutation as mentioned above, and this mutation will not improve fitness in the purple population. The conclusion is that the adaptive value of a mutation depends on *z*.

I can now bring together all of these elements to illustrate mathematically, just as Fisher did, and Orr in his review paper. The question is, what is the probability that a random mutation of effect size *r *is adaptive? This is annotated as *P*_{a}(*x*), the probability that a mutation of standardised effect size, *x*, is adaptive. The probability is drawn from a cumulative random normal distribution and subtracted from 1 [1-Φ(*x*)]. To get to the standardised effect size draw on the points I’ve discussed above, it is *x = **r √n / (2z) *which in words is the square root of the number of characters divided by twice the distance between the population and the adaptive peak, multiplied by the effect size. The result is that increases in the values of *n *and *r *, the complexity underlying adaptedness and effect size of the mutation respectively, will reduce the probability that a mutation is beneficial to the organism. Remember, larger effects are less likely to be able to land closer to the adaptive peak because they can overshoot. Further reductions in the probability of a mutation being adaptive come when *z*, the distance to the adaptive peak, is low. Once again this makes it less likely a new mutation can land in a more adaptive position. Fisher and Orr both sum this up with a simple graph showing the standardised effect size explaining the probability of it being beneficial. It can be recreated in R with this little snippet of script: *plot(function(x) 1-pnorm(x),0,3)*.

And you can play around with various values of *n*, *r*, and *z *using this function in R:* FGM = function (n,r,z){ x = r*sqrt(n)/(2*z); Px = 1-pnorm(x); }; Px = FGM(n= ,z= ,r= ); Px #* (set your own values of *n*, *r*, and *z*).

To conclude, it should now be clear that complexity underlying adaptation decreases the rate of evolution so more complex species should adapt more slowly than, adaptation will slow as a species nears it’s adaptive peak, and large effect mutations are unlikely to be adaptive. Given that the process of adaptation has been occurring for billions of years, and that selection is unlikely to cause large violent shifts in the adaptive peak it would seem safe to assume that most species are relatively near their adaptive peak which further reinforces the importance of small effect mutations, as stated by Orr; “precise adaptation is possible only if organisms can come to fit their environments by many minute adjustments.” (Note: large effect mutations could, with a big chunk of luck, land very close to an adaptive peak but this does require a species to start a long way from the peak).

Overall I think that, although Fisher’s Geometric Model is not the main focus of the paper, the Orr paper is an excellent way to introduce yourself to the theory. With a little time, extra reading, and playing in R it can give a very good understanding of the model. This understanding is certainly aided by the papers thorough discussion of paradigm shifts in adaptation theory. One important consideration is the role of QTL studies, looking for large effect mutations explaining adaptation rarely yields results of overwhelming proportions. Most, or possibly even all, QTL studies reveal that a substantial portion of the variance in traits is explained by many small effect loci, those that go undetected by current analytical methods. That’s not to say QTL studies have no value, but what is needed is improvement of methods that can detect small effect loci which is not going to be an easy task, and new acceptance that small effect loci play a significant collective roll and therefore theory and research effort should not ignore them.