Fitness landscapes and positional inheritance

Fitness landscapes are a common way of visualizing the relationship between variables associated with an organism and fitness. The height of the landscape represents fitness and the domain over which the fitness landscape is defined is composed of other variables associated with the organism. Most frequently the information associated with heritable elements is used - but other variables affecting fitness could be included as well - such as environmental factors.

In this post, we will apply the idea of fitness landscapes to simple systems involving positional inheritance. Hopefully this will help to illustrate how the concept of 'fitness' applies to these kinds of system. To create a plot of fitness, we have to say what we mean by fitness. Fitness is a notoriously overloaded and slippery idea in biology - as was once explained in a book chapter titled "An Agony in Five Fits". Here, we won't proscribe any particular definition of fitness, but rather will show how to apply some common definitions of fitness.

The first aspect of measuring fitness is to define what entities you are measuring the fitness of. If there are multiple types of organism in a system, you have to say which one you are interested in tracking the fitness of. In simple systems involving positional inheritance this decision is often relatively simple: since there's only one main candidate entity. For example, with lightning strikes tracking the reproducing tips of the lightning are the obvious candidate. With stream systems, the branching tips of the streams themselves would be the most obvious object of study. With propagating cracks, the crack tips would be the object of study. With diffusion-limited aggregation, the available aggregation points would be what was tracked. In many of these cases, the precision of the available measuring instruments may be a factor in deciding exactly what entities are tracked.

Having selected the entities to be measured, the next thing to do is to decide how to measure fitness. Although there are many fitness metrics used for different purposes, we can categorize them in a few main ways. Fitness metrics can be:

• Relative or absolute - depending on whether you are interested in relative success or absolute results;
• Expected or actual - actual fitness measure growth rates while expected fitness can be calculated in advance;
• Short or long term - the time horizon affects fitness measurements: offspring don't always result in grandchildren;
• Generational or per unit of time - measuring growth in generational time units can sometime be useful.

These all apply to positional inheritance systems - though generational measures of fitness are not much use there. It's like using generational fitness metrics with bacteria. Bacteria are mortal and do have lifespans - but their lifespans are not much like the human three score years and ten. It's more a case that the bacteria live until they have a fatal accident - and the frequency of bacterial deaths are heavily determined by environmental factors. Thus, the lifespan of a bacterium is not usually a particularly useful or interesting figure. Using generational fitness with simple positional inheritance systems is a lot like this.

A fitness landscape is usually a plot of fitness over gene space. The peaks illustrate where well-adapted organisms are likely to be found. The roughness of the fitness landscape influences whether and how quickly evolving organisms will be able to find the peaks.

With simple positional inheritance systems, the 'genes' in question are positions - since position is one of the main things that is inherited in these systems. So the domain of the fitness landscape plot is usually simple two or three dimensional space. Fitness measures how likely branching or splitting is to take place at points in that space. Since reproduction typically requires resources, fitness can be reasonably expected to be correlated with resource availability.

It is common for fitness landscapes to change over time. As the environment changes, different genetic combinations are favored - and the fitness landscape shifts dynamically. With simple positional inheritance systems fitness landscapes tend to change in a predictable manner - the highest peaks tend to be systematically eroded. Because reproduction requires resources, takes place where resources are plentiful and depletes local resources, resource-rich areas will be systematically exploited and eliminated.

Fitness landscapes only track the parameters specified in their domain. If other factors affect fitness, the calculated fitnesses will not be accurate if these are omitted. For example, it is common to leave environmental factors out of fitness landscape plots.This can result in a lack of realism. With simple positional inheritance systems, fitness can also depend on more than positional factors. For example, consider a spreading fire. The reproduction rate of flames will be heavily influenced by positional factors - such as the local availability of combustible material. However other factors can also affect the rate of flame reproduction - such as the wind direction and the temperature - these are often a function of time as well as position. If there are more factors you can add them to the domain of the fitness landscape - but then you get a more complex plot in a higher dimensional space - which might not be so easy to make use of.

References

• Tyler, Tim (2012) Positional inheritance
• Tyler, Tim (2012) Positional inheritance (video)