I’ll keep the same here as for example #01, Specs in Isle, and velo, the gene determining Specs’ velocity, and the same clonal population at time T0.
Let’s consider a particularity of gene velo. If two specific mutations are present on the same molecule, velo-d1 and velo-d2, they act synergistically and the velocity of the individual is Vmax. velo-d1 and velo-d2 independently don’t affect the velocity trait, thus they aren’t selected, and they aren’t punctual mutations.
Say that this occurs at time Td1-d2, and this is in a stage of Specs history that gives it a clear advantage, substantially increasing the movements of carrier Specs, this particular combination will spread rapidly in Specs population, as the carriers have plenty of time available for sex.
The history of Specs will be determined from the occurrence of this particular random event. It represents a jump in the evolution of the particular phenotypic trait, velocity.
A jump which amplitude will be inversely proportional to Td1/d2-T0.
If you observe the increase of velocity of the Specs and don’t know anything about their genetics, you may have the feeling that there is some Unknown Factor making them faster and faster. And that at time Td1/d2 another Unknown Factor changed the rules and produced the jump of the velocity’s increace. Then you have two solutions: considering some Mysterious Internal Blueprint due to Supernatural Causation being the cause of the velocity’s increase and some Mysterious Change of the Internal Blueprint at time Td1/d2, or inventing genetics and understand how Specs evolved from Spec SlowMotion (those Specs present at T0) to Spec AsFastAsPossible (those Specs living at time Ts, including the velocity’s sudden increase at Td1/d2.
And this is a third observation: Phenotypical traits may evolve in a directional way and by jumps, while mutations are still perfectly random.
But a jump at Vmax isn’t very probable. Let’s say that this is a smaller jump, an increase of x% of the initial velocity, V0, and this particular event I’ll call a velo-jump from now on; the plot of velocity versus time (generations) will be something like that :
Concommitant to the velocity’s increase there is a drop of the selection pressure (they are correlated) and then the evolution of the phenotype will follow the same path as in example 1, reaching a plateau.
The coexistence of the two mutations in a single velo molecule is a random phenomenon. So it’s impossible to say « when » and « if » it will happen during the Specs history. I run a few simulations, and I had a very pleasant surprise; a velo-jump above Vmax. I wasn’t expecting to talk here about the reversibility of the model, but here is what I got at the second run of the model:
There are two graphs in fig above. Both represent the same data, the one at the right isn’t versus time, but log(time), this is useful to see more clearly early events. The blue curve represent velocity and the red ones selection pressure. The genotype d1-d2 appears relatively late and the velocity gain set Vs above Vmax. This phenotype isn’t privileged, as in the case where food’s density increased, I presented in example 1. So it will be progressively lost, either by reversion or by apparition of new mutations. I was glad to see that « happen », the model used for simulations seems to be well constructed.
Reversibility is conserved.
I gave it a few more runs and I overlaid a series of graphs to show alternative Specs histories:
There are three series presented here, a, b and c, corresponding to velo-jumps of 10, 20 and 40% of V0 respectively. For each series I show graphs with the x axis as time or log(time). The red arrows at « series c », points to a story where the d1-d2 genotype didn’t appeared. The duration of Specs history is normalized here. I’ll come back to this.
While Vmax is reached for every run, which mean that the Specs will be optimized for their environment, the path, trough which this is achieved, is as random as the occurrence of the d1-d2 genotype; it may even not contain this particular event.
Now, why did I normalized the histories lengths? It’s only a matter of better visualisation. Let’s see at what generation d1-d2 appeared for a set of 1000 runs :
Quite random, ranging from 5 to 53777. To make it visible, without making a particular choice, the simplest way was to normalize the duration of the stories.
One may evaluate odds that d1-d2 appears during the Specs history; this is a simple probability calculation, predicting that d1-d2 may appear. If the probabilities are equal for both mutations, and p is the probability for each one, then the probability to get both simultaneously is pd1-d2 = p2.
It doesn’t say when.
On the other hand, you may evaluate what is the probability for d1-d2 to appear after n generation and that will be n * pd1-d2 = n* p2.
Randomness is preserved at the mutations level, and thus at the genotype level, the phenotype evolves directionally as long as it doesn’t exceed the optimal value (otherwise the direction change), and a particular genetic event may produce jumps of the phenotypic trait.
One couldn’t predict the history of the Specs otherwise then in terms of probabilities; the single predetermined element is Vmax, and this is only to facilitate calculations, the real value being a function of the Specs physiology and foods density.
The probability to have two identical histories (runs of the simulations) is low. Some cases are present (difficult to see on the graph, but there are 42 repetitions! Now, that the genotype d1-d2 occurs at generation g twice (or more) that doesn’t mean that the paths where identical!