## Part 2: The Mapping

### An Odd Machine

7 ml

23 ml

35 ml

3 ml

90 ml

2 ml

28 ml

85 ml

**mapped**

*from*the switch space

*to*the 1D whiskey space. It isn't important that it is a three-dimensional discrete system mapping to one-dimensional continuous system - we could replace the three switches with a continuous slider and map from that space to the whiskey drink space, and if my machine is capable of spitting out not just whiskey but whiskey mixed with cola, we could map from the space defined by the controls to a 2D whiskey drink space, like so:

*from one space to some other space*.

### The Genotype-Phenotype Map

In the framework we've been using here, this mysterious process can be captured in a way by what is called

**the genotype-phenotype map**. The idea is that if we have a space of possible genotypes we can ask for each possible genotype, what phenotype is produced? Biologists will often call this process of getting from genotype to phenotype

*development*. If we're considering the whole system, this is a mapping from a very high dimensional genotype space to a very high dimensional phenotype space. If, however, we are only interested in a part of the phenotype, for example the antibiotic resistance of a certain bacteria, then the mapping could be from a high dimensional subspace of genotype space to a one dimensional subspace of phenotype space.

I should note that development is also affected by the organism's environment, so the mapping from genotype space to phenotype space could change if the environment changes - but this is something we will (conveniently) ignore for now.

### Fitness

### Adding the Map Dimension

Gold stars and jolly ranchers for our past selves! And so we get this:

**added an extra dimension for the mapping**. We've added the space of possibilities that we're mapping to (in this case y-values from negative infinity to positive infinity) and plotted points or a line in our new space to represent the mapping.

Importantly, notice that we are mapping to a one-dimensional space (the space of possibilities for the variable y), so it is easy to add the additional dimension and draw the line. Will it still work if we're mapping to a space with more than one dimension? I don't see why not, but in most cases, we are concerned with mapping to one-dimensional spaces.

Could we map from a two-dimensional space to a one-dimensional space? Now our high-school selves are quietly writing down in their notebooks (it's not cool to raise your hand anymore) "z = x + y". You can think of spreading out the possibilities for x and y along two axes and then adding another dimension (axis) for the mapping to z. The result is a "surface" in three-dimensional space.

### And Finally: The Landscape

**landscape metaphor**, first introduced by Sewall Wright in 1932. Wright presented a metaphor in which each point in the space of combinations of alleles at different loci (similar to genotype space) was like a point on a physical landscape, and elevation represented the fitness of each genotype. Wright introduced the metaphor as a way of thinking about change in gene frequencies in populations of animals and as a way of thinking about how gene interactions affect long-term evolutionary dynamics (using the metaphor for both of these purposes has caused a load of confusion, see for example the section "Wrinkles in the Surface of Selective Value" in Provine's biography of Wright).

In the dialogue, the Tortoise and Achilles traverse a landscape. Imagine for a moment that the world is flat, and we are up in a satellite far above the two explorers. We could record the possible positions of the animals in two-dimensional space using a north-south axis and an east-west axis (please ignore the curvature of the earth). This is our space of possibilities. Now we'll add the mapping: for each possible position we want to know the elevation. Then we'll take this elevation mapping and add it to our space of possibilities as an extra dimension. And voila - we have a two dimensional space with a third dimension for the mapping, and the surface we've created is a representation of the actual surface of the physical landscape!

Here are the analogies of the

**traditional adaptive landscape**concept:

The Analogies:

**Genotype or Phenotype Space ----------- A Two-Dimensional Space of Possible Positions on a Landscape**

Fitness ------------------------------------------ The Elevation

Fitness ------------------------------------------ The Elevation

What I hope has come through in this theoretical introduction is a really slight change in thinking towards

**something more general**. Here it is:

The Analogies:

**Genotype or Phenotype Space ------ Any Space of Possibilities**

Fitness ------------------------------------- Any Mapping

Fitness ------------------------------------- Any Mapping

From my examples of mixed drinks and light switches, you can see that this is an incredibly broad analogy. The specific idea of a three-dimensional landscape is one example of many that fits within this structure.

**What makes it an important example is that it is where our terminology and our low-dimensional visualization come from**. We talk about

*peaks, valleys, ruggedness,*and

*hill-climbing,*and we picture

*rolling hills, ridges, cliffs,*and the like when we theorize.

I think this second interpretation is useful for two reasons. First, by beginning from a more general standpoint, we are forced to ask questions about what the space of possibilities and the mapping look like. This is a process of critical analogy. I mentioned that we use the physical landscape for our terminology and intuitive understanding of this kind of structure. However when we think about the adaptive landscape, we have to avoid letting this terminology and intuition from low-dimensional, continuous space mislead us when thinking about high-dimensional or discrete systems. We can avoid these pitfalls by being critical and asking questions as we use the metaphor, which comes more naturally when using the broader view.

Second, I think that putting many systems of change into this broad analogy and asking critical questions about how these systems are similar or different could yield considerable insights. The adaptive landscape metaphor in its strict form has provided biologists with an intuitive way to think of evolution for 80 years, playing a role in theory formation many times over the years. I think that we should encourage critical analogy not just with the physical landscape but also with systems from many disciplines.

My goal here was to introduce this concept in a general way, with an emphasis on a sweeping, interdisciplinary analogy that we use critically to understand changing systems throughout our world. In the next section we will dive into the technical dirt and start looking at some of the difficulties in talking about distance and moving around the landscape, and how these difficulties relate to very real problems in our scientific understanding of the mappings between genotype, phenotype, and fitness.